Solved: Correcting a Navigational Error A sailboat leaves

In a right triangle, if the length of the hypotenuse is 5 and the length of one of the other sides is 3, what is the length of the third side? (pp. 30-31).

True or False sin 52 = cos 48. (pp. 523-525)

If e is an acute angle, solve the equation tan e = . Express your answer in degrees, rounded to one decimal place. (pp. 649-653)

If e is an acute angle, solve the equation sin e = . (pp. 649-653)

True or False In a right triangle, one of the angles is 90 and the sum of the other two angles is 90.

In navigation or surveying, the __ from a point to a point P equals the acute angle e between ray OP and the vertical line through 0, the north-south line.

True or False In a right triangle, if two sides are known, we can solve the triangle.

True or False In a right triangle, if we know the two acute angles, we can solve the triangle.

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. b = 5, B = 20; find a, c, and A

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. b = 4, B = 10; find a, c, and A

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. a = 6, B = 40; find b, c, and A

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. a = 7, B = 50; find b, c, and A

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. b = 4, A = 1 0; find a, c, and B

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. b = 6, A = 20; find a, c, and B

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. a = 5, A = 25; find b, c, and B

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. a = 6, A = 40; find b, c, and B

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. c = 9, B = 20; find b, a, and A

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. C = 10, A = 40; find b, a, and B

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. a = 5, b = 3; find c, A, and B

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. a = 2, b = 8; find c, A, and B

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. a = 2, C = 5; find b, A, and B

In Problems 9-22, use the right triangle shown below. Then, using the given information, solve the triangle. b = 4, c = 6; find a, A, and B

Geometry The hypotenuse of a right triangle is 5 inches. If one leg is 2 inches, find the degree measure of each angle.

Geometry The hypotenuse of a right triangle is 3 feet. If one leg is 1 foot, find the degree measure of each angle.

Finding the Angle of Elevation of the Sun At 10 AM on April 26, 2006, a building 300 feet high casts a shadow 50 feet long. What is the angle of elevation of the Sun?

Directing a Laser Beam A laser beam is to be directed through a small hole in the center of a circle of radius 10 feet. The origin of the beam is 35 feet from the circle (see the figure). At what angle of elevation should the beam be aimed to ensure that it goes through the hole? _ - - - - - ill (f\ P---- -------- 35 ft------1 Laser

Finding the Speed of a Truck A state trooper is hidden 30 feet from a highway. One second after a truck passes, the angle e between the highway and the line of observation from the patrol car to the truck is measured. See the illustration. (a) If the angle measures 15, how fast is the truck traveling? Express the answer in feet per second and in miles per hour. (b) If the angle measures 20, how fast is the truck traveling? Express the answer in feet per second and in miles per hour. (c) If the speed limit is 55 miles per hour and a speeding ticket is issued for speeds of 5 miles per hour or more over the limit, for what angles should the trooper issue a ticket?

Security A security camera in a neighborhood bank is mounted on a wall 9 feet above the floor. What angle of depression should be used if the camera is to be directed to a spot 6 feet above the floor and 12 feet from the wall?

Finding the Bearing of an Aircraft A DC-9 aircraft leaves Midway Airport from runway 4 RIGHT, whose bearing is N400E. After flying for mile, the pilot requests permission to turn 90 and head toward the southeast. The permission is granted. After the airplane goes 1 mile in this direction, what bearing should the control tower use to locate the aircraft?

Finding the Bearing of a Ship A ship leaves the port of Miami with a bearing of S80E and a speed of 15 knots. After 1 hour, the ship turns 90 toward the south. After 2 hours, maintaining the same speed, what is the bearing to the ship from port?

Niagara Falls Incline Railway Situated between Portage Road and the Niagara Parkway directly across from the Canadian Horseshoe Falls, the Falls Incline Railway is a funicular that carries passengers up an embankment to Table Rock Observation Point. If the length of the track is 51.8 meters and the angle of inclination is 362', determine the height of the embankment. Source: www.niagaraparks.com

Sears Tower The Sears Tower in Chicago is the third tallest building in the world and is topped by a high antenna. A surveyor on the ground makes the following measurement: 1. The angle of elevation from his position to the top of the building is 34. 2. TIle distance from his position to the top of the building is 2593 feet. 3. The distance from his position to the top of the antenna is 2743 feet. (a) How far away from the (base of the) building is the surveyor located? (b) How tall is the building? (c) What is the angle of elevation from the surveyor to the top of the antenna? (d) How tall is the antenna? Source: www.infoplease.comlce6IusIA0844218.html

Chicago Skyscrapers The angle of inclination from the base of the John Hancock Center to the top of the main structure of the Sears Tower is approximately 10.3. If the main structure of the Sears Tower is 1451 feet tall, how far apart are the two skyscrapers? Assume the bases of the two buildings are at the same elevation. Source: www.emporis.com

Estimating the Width of the Mississippi River A tourist at the top of the Gateway Arch (height, 630 feet) in St. Louis, Missouri observes a boat moored on the Illinois side of the Mississippi River 2070 feet directly across from the Arch. She also observes a boat moored on the Missouri side T 630 1 11-'(f--,----- MO IL 2070 ft -------J.I directly across from the first boat (see diagram). Given that B = cot-1 , estimate the width of the Mississippi River at the St. Louis riverfront. Source: u.s. Army Corps of Engineers

Finding the Pitch of a Roof A carpenter is preparing to put a roof on a garage that is 20 feet by 40 feet by 20 feet. A steel support beam 46 feet in length is positioned in the center of the garage. To support the roof, another beam will be attached to the top of the center beam (see the figure). At what angle of elevation is the new beam? In other words, what is the pitch of the roof? T 20 ft 1 New beam

Shooting Free Throws in Basketball The eyes of a basketball player are 6 feet above the floor. TIle player is at the freethrow line, which is 15 feet from the center of the basket rim (see the figure). What is the angle of elevation from the player's eyes to the center of the rim? [Hint: The rim is 10 feet above the floor.] I - - - I _ _ ______ I _-___ :. :: 15 ft 10ft 1

Geometry Find the value of the angle e in degrees round ed to the nearest tenth of a degree.

Surveillance Satellites A surveillance satellite circles Earth at a height of h miles above the surface. Suppose that d is the distance, in miles, on the surface of Earth that can be observed from the satellite. See the illustration. (a) Find an equation that relates the central angle e to the height h. (b) Find an equation that relates the observable distance d and e. (c) Find an equation that relates d and h. (d) If d is to be 2500 miles, how high must the satellite orbit above Earth? (e) If the satellite orbits at a height of 300 miles, what distance d on the surface can be observed?

The Gibb's Hill Lighthouse, Southampton, Bermuda In operation since 1846, the Gibb's Hill Lighthouse stands 117 feet high on a hill 245 feet high, so its beam of light is 362 feet above sea level. A brochure states that ships 40 miles away can see the light and planes flying at 10,000 feet can see it 120 miles away. Verify the accuracy of these statements. What assumption did the brochure make about the height of the ship?

The difference formula for the sine function is sin(A - B) = __ . (p. 630)

If e is an acute angle, solve the equation cos e = V;. (pp. 649-653)

The two triangles shown are similar. Find the missing length. (pp. 30-35)

If none of the angles of a triangle is a right angle, the triangle is called __ .

For a triangle with sides a, b, c and opposite angles A, B, e, the Law of Sines states that ____.

True or False An oblique triangle in which two sides and an angle are given always results in at least one triangle.

True or False The sum of the angles of any triangle equals 180.

True or False The ambiguous case refers to the fact that, when two sides and the angle opposite one of them are given, sometimes the Law of Sines cannot be used.

In Problems 9-16, solve each triangle. A. 5

In Problems 9-16, solve each triangle. 4

In Problems 9-16, solve each triangle. B c

In Problems 9-16, solve each triangle. 10 A c

In Problems 9-16, solve each triangle. c

In Problems 9-16, solve each triangle. 5 5 b :

In Problems 9-16, solve each triangle. c

In Problems 9-16, solve each triangle. c

In Problems 17-24, solve each triangle. A = 40, B = 20, a = 2

In Problems 17-24, solve each triangle. A = 50, e = 20, a = 3

In Problems 17-24, solve each triangle. B = 70, e = 10, b = 5

In Problems 17-24, solve each triangle. A = 70, B = 60, c = 4

In Problems 17-24, solve each triangle. A = 1 10, e = 30, c = 3

In Problems 17-24, solve each triangle. B = 10, e = 100, b = 2

In Problems 17-24, solve each triangle. A = 40, B = 40, c = 2

In Problems 17-24, solve each triangle. B = 20, e = 70, a = 1

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. a = 3, b = 2, A = 50

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. b = 4, c = 3, B = 40

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. b = 5, c = 3, B = 100

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. a = 2, c = 1, A = 120

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. a = 4, b = 5, A = 60

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. b = 2, c = 3, B = 40

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. b = 4, c = 6, B = 20

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. a = 3, b = 7, A = 70

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. a = 2, c = 1, e = 100

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. b = 4, c = 5, B = 95

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. a = 2, c = 1, e = 25

In Problems 25-36, two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results. b = 4, c = 5, B = 40

Rescue at Sea Coast Guard Station Able is located 150 miles due south of Station Baker. A ship at sea sends an SOS call that is received by each station. The call to Station Able indicates that the ship is located N55E; the call to Station Baker indicates that the ship is located S600E. (a) How far is each station from the ship? (b) If a helicopter capable of flying 200 miles per hour is dispatched from the station nearest the ship, how long will it take to reach the ship?

Surveying Consult the figure to the right. To find the distance from the house at A to the house at B, a surveyor measures L. RAe to be 35 and then walks off a distance of 100 feet to C and measures L ACB to be 50. What is the distance from A to B?

Finding the Length of a Ski Lift Consul t the figure. To find the length of the span of a proposed ski lift from P ta Q, a surveyor measures L D PQ to be 25 and then walks off a distance of 1000 feet to R and measures L PRQ to be 15. What is the distance from P ta Q? Q R 1-1000 It --l

Finding the Height of a Mountain Use the illustration in Problem 39 t o find the height QD of the mountain.

Finding the Height of an Airplane An aircraft is spotted by two observers who are 1000 feet apart. As the airplane passes over the line joining them, each observer takes a sighting of the angle of elevation to the plane, as indicated in the figure. How high is the airplane? ; ; ; ; &;" 40 p ; ; ; ; 1 000 It

Finding the Height of the Bridge over the Royal Gorge The highest bridge in the world is the bridge over the Royal Gorge of the Arkansas River in Colorado. Sightings to the same point at water level directly under the bridge are taken from each side of the 880-foot-long bridge, as indicated in the figure. How high is the bridge? Source: Guinness Book of World Records

Landscaping Pat needs to determine the height of a tree before cutting it down to be sure that it will not faLl on a nearby fence. The angle of elevation of the tree from one position on a flat path from the tree is 30, and from a second position 40 feet farther along this path it is 20. What is the height of the tree?

Construction A loading ramp 10 feet long that makes an angle of 18 with the horizontal is to be replaced by one that makes an angle of 12 with the horizontal. How long is the new ramp?

Commercial Navigation Adam must fly home to St. Louis from a business meeting in Oklahoma City. One flight option flies directly to St. Louis, a distance of about 461.1 miles. A second flight option flies first to Kansas City and then connects to St. Louis. The bearing from Oklahoma City to Kansas City is N29.6E, and the bearing from Oklahoma City to St. Louis is N57.7E. The bearing from St. Louis to Oklahoma City is S62.1W, and the bearing from St. Louis to Kansas City is N79.4W. How many more frequent flyer miles will Adam receive if he takes the connecting flight rather than the direct flight?

Time Lost due to a Navigation Error In attempting to fly from city P to city Q, an aircraft followed a course that was 10 in error, as indicated in the figure. After flying a distance of 50 miles, the pilot corrected the course by turning at point R and flying 70 miles farther. If the constant speed of the aircraft was 250 miles per hour, how much time was lost due to the error?

Finding the Lean of the Leaning Tower of Pisa The famous Leaning Tower of Pisa was originally 184.5 feet high.* At a distance of 123 feet from the base of the tower, the angle of elevation to the top of the tower is found to be 60. Find L. RPQ indicated in the figure. Also, find the perpendicular distance from R to PQ.184.5 It R I . 1:1,\ . , \ ,- .\ , \ , \ , \ '"", \ , \ , \ , \ , \ J \ , \ , \ , \ , \ f \ , 60C 123 It Q

Crankshafts on Cars On a certain automobile, the crankshaft is 3 inches long and the connecting rod is 9 inches long (see the figure). At the time when L. OPQ is 15, how far is the piston (P) from the center (0) of the crankshaft?

Constructing a Highway U.S. 41, a highway whose primary directions are north-south, is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of highway needed to go around the bay? Ocean

Calculating Distances at Sea The navigator of a ship at sea spots two lighthouses that she knows to be 3 miles apart along a straight seashore. She determines that the angles formed between two line-of-sight observations of the lighthouses and the line from the ship directly to shore are 15 and 35. See the illustration. (a) How far is the ship from lighthouse P? (b) How far is the ship from lighthouse Q? (c) How far is the ship from shore?

Designing an Awning An awning that covers a sliding glass door that is 88 inches tall forms an angle of 50 with the wall. The purpose of the awning is to prevent sunlight from entering the house when the angle of elevation of the Sun is more than 65. See the figure. Find the length L of the awning. 88" l?::J 1 I I I I I I I I I I I W t::. ----- Step

Finding Distances A forest ranger is walking on a path inclined at 5 to the horizontal directly toward a 100-foot-tall fire observation tower. The angle of elevation from the path to the top of the tower is 40. How far is the ranger from the tower at this time? horizontal

Great Pyramid of Cheops One of the original Seven Wonders of the World, the Great Pyramid of Cheops was built about 2580 Be. Its original height was 480 feet 11 inches, but due to the loss of its topmost stones, it is now shorter. Find the current height of the Great Pyramid, using the information given in the illustration. Source: Guinness Book of World Records

Determining the Height of an Aircraft Two sensors are spaced 700 feet apart along the approach to a small airport. When an aircraft is nearing the airport, the angle of elevation from the first sensor to the aircraft is 20, and from the second sensor to the aircraft it is 15. Determine how high the aircraft is at this time.

Mercury The distance from the Sun to Earth is approximately 149,600,000 kilometers (km). The distance from the Sun to Mercury is approximately 57,910,000 km. The elongation angle a is the angle formed between the line of sight from Earth to the Sun and the line of sight from Earth to Mercury. See the figure. Suppose that the elongation angle for Mercury is 150 Use this information to find the possible distances between Earth and Mercury.

Venus The distance from the Sun to Earth is approximately 149,600,000 km. The distance from the Sun to Venus is approximately 108,200,000 km. The elongation angle a is the angle formed between the line of sight from Earth to the Sun and the line of sight from Earth to Venus. Suppose that the elongation angle for Venus is 100 Use this information to find the possible distances between Earth and Venus.

The Original Ferris Wheel George Washington Gale Ferris, Jr. designed the original Ferris wheel for the 1893 World's Columbian Exposition in Chicago, Illinois. The wheel had 36 equally spaced cars each the size of a school bus. The distance between adjacent cars was approximately 22 feet. Determine the diameter of the wheel to the nearest foot. Source: Carnegie Library of Pittsburgh, www.clpgh.org

Mollweide's Formula For any triangle, MoUweide's Formula (named after Karl Mollweide, 1774-1825) states that a + b COSD(A -B)] c sinG C) Derive it. [Hint: Use the Law of Sines and then a Sum-to-Product Formula. Notice that this formula involves all six parts of a triangle. As a result, it is sometimes used to check the solution of a triangle.]

Mollweide's Formula Another form of Mollweide's Formula is a-b sin[ (A -B)] c cosGc) Derive it.

For any triangle, derive the formula a = b cos C + c cos B [Hint: Use the fact that sin A = sin(180 -B -C).]

Law of Tangents For any triangle, derive the Law of Tangents. a -b a + b tan[ (A -B) ] tan[ (A + B)] [Hint: Use Mollweide' s Formula.]

Circumscribing a Triangle Show that sin A a sin B sin C b c 1 2,. where r is the radius of the circle circumscribing the triangle PQR whose sides are a, b, and c, as shown in the figure. [Hint: Draw the diameter P P'. Then B = L. PQR = L. PP'R, and angle L. PRP' = 90.] Q P

Make up three problems involving oblique triangles. One should result in one triangle, the second in two triangles, and the third in no triangle.

What do you do first if you are asked to solve a triangle and are given one side and two angles?

What do you do first if you are asked to solve a triangle and are given two sides and the angle opposite one of them?

Write the formula for the distance d from PI = (XI, YI) to P2 = (X2' Y2)' (p. 157)

If () IS an acute angle, solve the equatIon cos () = --. (pp. 649-653)

If three sides of a triangle are given, the Law of ____ is used to solve the triangle.

If one side and two angles of a triangle are given, the Law of ____ is used to solve the triangle.

If two sides and the included angle of a triangle are given, the Law of ____ is used to solve the triangle.

True or False Given only the three sides of a triangle, there is insufficient information to solve the triangle.

True or False Given two sides and the included angle, the first thing to do to solve the triangle is to use the Law of Sines.

True or False A special case of the Law of Cosines is the Pythagorean Theorem.

In Problems 9-16, solve each triangle. 2 b 4

In Problems 9-16, solve each triangle. B 30 4

In Problems 9-16, solve each triangle.

In Problems 9-16, solve each triangle. 20 A 5

In Problems 9-16, solve each triangle. 5 A 8

In Problems 9-16, solve each triangle. d B A 4

In Problems 9-16, solve each triangle. 4

In Problems 9-16, solve each triangle. 4

In Problems 17-32, solve each triangle. a = 3, b = 4, C = 40

In Problems 17-32, solve each triangle. a = 2, c = 1, B = 10

In Problems 17-32, solve each triangle. b = 1, C = 3, A = 80

In Problems 17-32, solve each triangle. a = 6, b = 4, C = 60

In Problems 17-32, solve each triangle. a = 3, c = 2, B = 110

In Problems 17-32, solve each triangle. b = 4, C = 1, A = 120

In Problems 17-32, solve each triangle. a = 2, b = 2, C = 50

In Problems 17-32, solve each triangle. a = 3, c = 2, B = 90

In Problems 17-32, solve each triangle. a = 1 2, b = 13, C = 5

In Problems 17-32, solve each triangle. a = 4, b = 5, c = 3

In Problems 17-32, solve each triangle. a = 2, b = 2, c = 2

In Problems 17-32, solve each triangle. a = 3, b = 3, c = 2

In Problems 17-32, solve each triangle. a = 5, b = 8, c = 9

In Problems 17-32, solve each triangle. a = 4, b = 3, c = 6

In Problems 17-32, solve each triangle. a = 10, b = 8, c = 5

In Problems 17-32, solve each triangle. a = 9, b = 7, c = 10

Distance to the Green A golfer hits an errant tee shot that lands in the rough. A marker in the center of the fairway is 150 yards from the center of the green. While standing on the marker and facing the green, the golfer turns 110 toward his ball. He then paces off 35 yards to his ball. See the figure. How far is the ball from the center of the green? 150 yd \ \ \ \ \ \ \ \ \ \ \ \ \ _______ ball

Navigation An airplane flies due north from Ft. Myers to Sarasota, a distance of 150 miles, and then turns through an angle of 50 and flies to Orlando, a distance of 100 miles. See the figure on next page. 5 A B = 10 B = 110 B = 90 c = 2 c = 6 marker c 14 "d B A 4 19. b = 1, C = 3, A = 80 22. b = 4, C = 1, A = 120 25. a = 1 2, b = 13, C = 5 28. a = 3, b = 3, c = 2 31. a = 10, b = 8, c = 5 \ 150 yd \ \ \ \ \ \ \ \ \ \ \ \ \ _______ ball 150 mi Ft. Myers (a) How far is it directly from Ft. Myers to Orlando? (b) What bearing should the pilot use to fly directly from Ft. Myers to Orlando?

Avoiding a Tropical Storm A cruise ship maintains an average speed of 15 knots in going from San Juan, Puerto Rico, to Barbados, West Indies, a distance of 600 nautical miles. To avoid a tropical storm, the captain heads out of San Juan in a direction of 20 off a direct heading to Barbados. The captain maintains the IS-knot speed for 10 hours, after which time the path to Barbados becomes clear of storms. (a) Through what angle should the captain turn to head directly to Barbados? (b) Once the turn is made, how long will it be before the ship reaches Barbados if the same IS-knot speed is maintained?

Revising a Flight Plan In attempting to fly from Chicago to Louisville, a distance of 330 miles, a pilot inadvertently took a course that was 10 in error, as indicated in the figure. (a) If the aircraft maintains an average speed of 220 miles per hour and if the error in direction is discovered after " " " Louisville " " ---- " ------ Chicago Error detected here 15 minutes, through what angle should the pilot turn to head toward Louisville? (b) What new average speed should the pilot maintain so that the total time of the trip is 90 minutes?

Major League Baseball Field A Major League baseball diamond is actually a square 90 feet on a side. The pitching rubber is located 60.5 feet from home plate on a line joining home plate and second base. (a) How far is it from the pitching rubber to first base? (b) How far is it from the pitching rubber to second base? (c) If a pitcher faces home plate, through what angle does he need to turn to face first base?

Little League Baseball Field According to Little League baseball official regulations, the diamond is a square 60 feet on a side. The pitching rubber is located 46 feet from home plate on a line joining home plate and second base. (a) How far is it from the pitching rubber to first base? (b) How far is it from the pitching rubber to second base? (c) If a pitcher faces home plate, through what angle does he need to turn to face first base?

Finding the Length of a Guy Wire The height of a radio tower is 500 feet, and the ground on one side of the tower slopes upward at an angle of 10 (see the figure). (a) How long should a guy wire be if it is to connect to the top of the tower and be secured at a point on the sloped side 100 feet from the base of the tower? (b) How long should a second guy wire be if it is to connect to the middle of the tower and be secUl'ed at a point 100 feet from the base on the flat side? 500 ft I I I I I I ,'100 ft

Finding the Length of a Guy Wire See the figure below. A radio tower 500 feet high is located on the side of a hill with an inclination to the horizontal of S. How long should two guy wires be if they are to connect to the top of the tower and be secured at two points 100 feet directly above and directly below the base of the tower?

Wrigley Field, Home of tbe Chicago Cubs The distance from home plate to the fence in dead center in Wrigley Field is 400 feet (see the figure). How far is it from the fence in dead center to thi l"d base?

Little League Baseball The distance from home plate to the fence in dead center at the Oak Lawn Little League field is 280 feet. How far is it from the fence in dead center to third base? [Hint: The distance between the bases in Little League is 60 feet.]

Building a Swing Set Clint is building a wooden swing set for his children. Each supporting end of the swing set is to be an A-frame constructed with two 10-foot-long 4 by 4s joined at a 45 angle. To prevent the swing set from tipping over, Clint wants to secure the base of each A-frame to concrete footings. How far apart should the footings for each A-frame be?

Rods and Pistons Rod OA rotates about the fixed point 0 so that point A travels on a circle of radius r. Connected to point A is another rod AB of length L > 2,. , and point B is connected to a piston. See the figure. Show that the distance x between point 0 and point B is given by x = ,. cos (j + yr2 cos2 (j + L 2 - r2 where (j is the angle of rotation of rod A.

Geometry Show that the length d of a chord of a circle of radius r is given by the formula d . (j = 2r Sll1 - 2 where (j is the central angle formed by the radii to the ends of the chord. See the figure. Use this result to derive the fact that sin (j < (j, where (j > 0 is measured in radians.

For any triangle, show that cos f = 2 "V 1 where s = 2 (a + b + c). [Hint: Use a Half-angle Formula and the Law of Cosines.]

For any triangle show that . C )r:"( - a-:--) (: - s- -b) Sln - = 2 ab 1 wheres = 2(a + b + c).

Use the Law of Cosines to prove the identity --cos A cos B cos C a2 + b2 + c2 + --+ --= ----- abc

What do you do first if you are asked to solve a triangle and are given two sides and the included angle?

What do you do first if you are asked to solve a triangle and are given three sides?

Make up an applied problem that requires using the Law of Cosines.

Write down your strategy for solving an oblique triangle.

The area of a triangle whose base is b and whose height is h is __ (p. 31)

If three sides of a triangle are given, ___ Formula is used to find the area of the triangle.

True or False No formula exists for finding the area of a triangle when only three sides are given.

True or False Given two sides and the included angle, there is a formula that can be used to find the area of the triangle.

In Problems 5-12, find the area of each triangle. Round answers to two decimal places. 2 b 4

In Problems 5-12, find the area of each triangle. Round answers to two decimal places. BW a 3 4

In Problems 5-12, find the area of each triangle. Round answers to two decimal places. 3 c

In Problems 5-12, find the area of each triangle. Round answers to two decimal places. -------- 2 ------ 5

In Problems 5-12, find the area of each triangle. Round answers to two decimal places. 6 5 8

In Problems 5-12, find the area of each triangle. Round answers to two decimal places. 8 5 4

In Problems 5-12, find the area of each triangle. Round answers to two decimal places. 4

In Problems 5-12, find the area of each triangle. Round answers to two decimal places. 4

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. a = 3, b = 4, C = 40

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. a = 2, c = 1, B = 100

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. b = 1, c = 3, A = 800

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. a = 6, b = 4, C = 600

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. a = 3, c = 2, B = 1 10

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. b = 4, c = 1, A = 120

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. a = 12, b = 1 3, c = 5

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. a = 4, b = 5, c = 3

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. a = 2, b = 2, c = 2

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. a = 3, b = 3, c = 2

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. a = 5, b = 8, c = 9

In Problems 13-24, find the area of each triangle. Round answers to two decimal places. a = 4, b = 3, c = 6

Area of an ASA Triangle If two angles and the included side are given, the third angle is easy to find. Use the Law of Sines to show that the area K of a triangle with side a and angles A, B, and C is a 2 sin B sin C K =----- 2 sin A

Area of a Triangle Prove the two other forms of the formula given in Problem 25. b 2 sin A sin C c 2 sin A sin B K = and K = :.....---- 2 sin B 2 sin C

In Problems 27-32, use the results of Problem 25 or 26 to find the area of each triangle. Round answers to two decimal places. A = 40 , B = 20 , a = 2

In Problems 27-32, use the results of Problem 25 or 26 to find the area of each triangle. Round answers to two decimal places. A = 50 , C = 20, a = 3

In Problems 27-32, use the results of Problem 25 or 26 to find the area of each triangle. Round answers to two decimal places. B = 70 , C = 10, b = 5

In Problems 27-32, use the results of Problem 25 or 26 to find the area of each triangle. Round answers to two decimal places. A = 70, B = 60, c = 4

In Problems 27-32, use the results of Problem 25 or 26 to find the area of each triangle. Round answers to two decimal places. A = 110, B = 30, c = 3

In Problems 27-32, use the results of Problem 25 or 26 to find the area of each triangle. Round answers to two decimal places. B = 10, C = 100, b = 2

Area of a Segment Find the area of the segment (shaded in blue in the figure) of a circle whose radius is 8 feet, formed by a central angle of 70. [Hint: Subtract the area of the triangle from the area of the sector to obtain the area of the segment.]

Area of a Segment Find the area of the segment of a circle whose radius is 5 inches, formed by a central angle of 40.

Cost of a Triangular Lot The dimensions of a triangular lot are 100 feet by 50 feet by 75 feet. If the price of such land is $3 per square foot, how much does the lot cost?

Amount of Material to Make a Tent A cone-shaped tent is made from a circular piece of canvas 24 feet in diameter by removing a sector with central angle 100 and connecting the ends. What is the surface area of the tent?

Dimensions of Home Plate The dimensions of home plate at any major league baseball stadium are shown. Find the area of home plate. 8.5 in. 8.5 in. 17 in.

Computing Areas Find the area of the shaded region enclosed in a semicircle of diameter 10 inches. The length of the chord PQ is 8 inches. [Hint: Triangle PQR is a right triangle.]

Geometry Consult the figure, which shows a circle of radius r with center at O. Find the area K of the shaded region as a function of the central angle e.

Approximating the Area of a Lake To approximate the area of a lake, a surveyor walks around the perimeter of the lake, taking the measurements shown in the illustration. Using this technique, what is the approximate area of the lake? [Hint: Use the Law of Cosines on the three triangles shown and then find the sum of their areas.]

The Flatiron Building Completed in 1902 in New York City, the Flatiron Building is triangular shaped and bounded by 22nd Street, Broadway, and 5th Avenue. The building measures approximately 87 feet on the 22nd Street side, 1 90 feet on the Broadway side, and 173 feet on the 5th Avenue side. Approximate the ground area covered by the building. Source: Sarah B radford Landau and Carl W. Condit, Rise of the New York Skyscraper: 1865-1913. New Haven, CT: Yale University Press, 1 996

Bermuda Triangle The Bermuda Triangle is roughly defined by Hamilton, Bermuda; San Juan, Puerto Rico; and Fort Lauderdale, Florida. The distances from Hamilton to Fort Lauderdale, Fort Lauderdale to San Juan, and San Juan to Hamilton are approximately 1028, 1046, and 965 miles respectively. Ignoring the curvature of Earth, approximate the area of the Bermuda Triangle. Source: www. worldatlas. com

Geometry Refer to the figure. If 10AI = 1, show that: (a) Area AOAC = sin a cos a (b) Area AOCB = IOBI2 sin f3 cos f3 (c) Area AOAB = IOBI sin(a + (3) d lOBI = cos a ( ) cos f3 (e) sin(a + (3) = sin a cos f3 + cos a sin f3 [Hint: Area AOAB = Area AOAC + Area AOCB.] B -..LI---..I-I..A

Geometry Refer to the figure, in which a unit circle is drawn. The line segment DB is tangent to the circle and e is acute. (a) Express the area of AOBC in terms of sin e and cos e. (b) Express the area of AOBD in terms of sin e and cos e. (c) The area of the sector OBC of the circle is e, where e is measured in radians. Use the results of parts (a) and (b) and the fact that Area AOBC < Area OBC < Area AOBD to show that -1 e 1 1 < --sin e < -- cos e y 1 -1 x

The Cow Problem':' A cow is tethered to one corner of a square barn, 10 feet by 10 feet, with a rope 1 00 feet long. What is the maximum grazing area for the cow? [Hint: See the illustration.]

Another Cow Problem If the barn in Problem 45 is rectangular, 10 feet by 20 feet, what is the maximum grazing area for the cow?

Perfect Triangles A perfect triangle is one having natural number sides for which the area is numerically equal to the perimeter. Show that the triangles with the given side lengths are perfect. (a) 9, 10, 17 (b) 6, 25, 29 SOllrce: M.Y. Bonsangue, G. E. Gannon, E. Buchman, and N. Gross, "In Search of Perfect Triangles," Mathematics Teachel; Vol. 92, No. 1 , 1 999: 56-61

If hI, h2' and h3 are the altitudes dropped from P, Q, and R, respectively, in a triangle (see the figure), show that 1 lIs - + - + - = - h1 h2 h3 K R a where K is the area of the triangle and s = (a + b + c). 2K [Hint: h\ = -.Ja

Show that a formula for the altitude h from a vertex to the opposite side a of a triangle is h = a sin B sin C sin A

Inscribed Circle For Problems 50-53, Ihe lines (hal bisect each angle of a triangle mee( in a single point 0, and the perpendicular distance r from 0 to each side of (he triangle is (he same. The circle wilh center al 0 and radius r is called (he inscribed circle of the Iriangle (see Ihe figure). R Apply the formula from Problem 49 to triangle OPQ to show that . A . B C S1l1 2: S1l1 2 C cos 2

Inscribed Circle For Problems 50-53, Ihe lines (hal bisect each angle of a triangle mee( in a single point 0, and the perpendicular distance r from 0 to each side of (he triangle is (he same. The circle wilh center al 0 and radius r is called (he inscribed circle of the Iriangle (see Ihe figure). R Use the result of Problem 50 and the result of Problems 46 and 47 in Section 9.3 to show that C s - c cot - 2 = --r 1 where s = '2 ( a + b + c).

Inscribed Circle For Problems 50-53, Ihe lines (hal bisect each angle of a triangle mee( in a single point 0, and the perpendicular distance r from 0 to each side of (he triangle is (he same. The circle wilh center al 0 and radius r is called (he inscribed circle of the Iriangle (see Ihe figure). R Show that A B C s cot - + cot - + cot - = - 2 2 2 I'

Inscribed Circle For Problems 50-53, Ihe lines (hal bisect each angle of a triangle mee( in a single point 0, and the perpendicular distance r from 0 to each side of (he triangle is (he same. The circle wilh center al 0 and radius r is called (he inscribed circle of the Iriangle (see Ihe figure). R Show that the area K of triangle PQR is K = rs where s = '2 (a + b + c). Then show that r = )(s - a)(s b)(s - c)

What do you do first if you are asked to find the area of a triangle and are given two sides and the included angle?

What do you do first if you are asked to find the area of a triangle and are given three sides?

The amplitude A and period T of f(x) = 5 sin (4x) are __ and __ . (pp. 563-569)

The motion of an object obeys the equation d = 4 cos(6t). Such motion is described as ____ ____ . The number 4 is called the _____.

When a mass hanging from a spring is pulled down and then released, the motion is called ____ if there is no frictional force to retard the motion, and the motion is called ____ if there is friction.

True or False If the distance d of an object from its rest position at time t is given by a sinusoidal graph, the motion of the object is simple harmonic motion.

In Problems 5-8, an object attached to a coiled spring is pulled down a distance a fi'om its rest position and then released. Assuming that the motion is simple harmonic with period T, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up. a = 5; T = 2 seconds

In Problems 5-8, an object attached to a coiled spring is pulled down a distance a fi'om its rest position and then released. Assuming that the motion is simple harmonic with period T, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up. a = 10; T = 3 seconds

In Problems 5-8, an object attached to a coiled spring is pulled down a distance a fi'om its rest position and then released. Assuming that the motion is simple harmonic with period T, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up. a = 6; T = 7T' seconds

In Problems 5-8, an object attached to a coiled spring is pulled down a distance a fi'om its rest position and then released. Assuming that the motion is simple harmonic with period T, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up. a = 4; T = 2 seconds

Rework Problem 5 under the same conditions except that, at time t = 0, the object is at its resting position and moving down.

Rework Problem 6 under the same conditions except that, at time t = 0, the object is at its resting position and moving down.

Rework Problem 7 under the same conditions except that, at time t = 0, the object is at its resting position and moving down.

Rework Problem 8 under the same conditions except that, at time t = 0, the object is at its resting position and moving down.

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its resting position? (c) What is the time required for one oscillation? (d) What is the frequency? d = 5 sin(3t)

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its resting position? (c) What is the time required for one oscillation? (d) What is the frequency? d = 4 sin(2t)

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its resting position? (c) What is the time required for one oscillation? (d) What is the frequency? d = 6 cos(7T't)

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its resting position? (c) What is the time required for one oscillation? (d) What is the frequency? d = 5 cos(t)

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its resting position? (c) What is the time required for one oscillation? (d) What is the frequency? d = -3 sinGt)

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its resting position? (c) What is the time required for one oscillation? (d) What is the frequency? d = -2 cos(2t)

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its resting position? (c) What is the time required for one oscillation? (d) What is the frequency? d = 6 + 2 cos(27T't)

In Problems 13-20, the displacement d (in meters) of an object at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its resting position? (c) What is the time required for one oscillation? (d) What is the frequency? d = 4 + 3 sin(7T't)

In Problems 21-24, graph each damped vibration curve for 0 :S t :S 27 1. d(t) = e-I/-rr cos(2t )

In Problems 21-24, graph each damped vibration curve for 0 :S t :S 27 1. d(t) = e-I/2r. cos(2t)

In Problems 21-24, graph each damped vibration curve for 0 :S t :S 27 1. del) = e-I / 2 r. cos t

In Problems 21-24, graph each damped vibration curve for 0 :S t :S 27 1. d(t) = e-I/4-rr cos [

In Problems 25-32, use the method of adding y-coordinates to graph each function. f(x) = x + cos x

In Problems 25-32, use the method of adding y-coordinates to graph each function. f(x) = x + cos(2x)

In Problems 25-32, use the method of adding y-coordinates to graph each function. f(x) = x - sin x

In Problems 25-32, use the method of adding y-coordinates to graph each function. f ( x) = x - cos x

In Problems 25-32, use the method of adding y-coordinates to graph each function. f(x) = sin x + cos x

In Problems 25-32, use the method of adding y-coordinates to graph each function. f(x) = sin(2x) + cos x

In Problems 25-32, use the method of adding y-coordinates to graph each function. g(x) = sin x + sin(2x)

In Problems 25-32, use the method of adding y-coordinates to graph each function. g(x) = cos(2x) + cos x

In Problems 33-38, an object of mass m (in grams) altached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is T (in seconds) under simple harmonic motion. (a) Write an equation that relates the distance d of the object from its rest position after t seconds. (b) Graph the equation found in part (a) for 5 oscillations using a graphing utility. m = 25, a = 10, b = 0.7, T = 5

In Problems 33-38, an object of mass m (in grams) altached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is T (in seconds) under simple harmonic motion. (a) Write an equation that relates the distance d of the object from its rest position after t seconds. (b) Graph the equation found in part (a) for 5 oscillations using a graphing utility. m = 20, a = 15, b = 0.75, T = 6

In Problems 33-38, an object of mass m (in grams) altached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is T (in seconds) under simple harmonic motion. (a) Write an equation that relates the distance d of the object from its rest position after t seconds. (b) Graph the equation found in part (a) for 5 oscillations using a graphing utility. m = 30, a = 18, b = 0.6, T = 4

In Problems 33-38, an object of mass m (in grams) altached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is T (in seconds) under simple harmonic motion. (a) Write an equation that relates the distance d of the object from its rest position after t seconds. (b) Graph the equation found in part (a) for 5 oscillations using a graphing utility. m = 15, a = 16, b = 0.65, T = 5

In Problems 33-38, an object of mass m (in grams) altached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is T (in seconds) under simple harmonic motion. (a) Write an equation that relates the distance d of the object from its rest position after t seconds. (b) Graph the equation found in part (a) for 5 oscillations using a graphing utility. m = 10, a = 5, b = 0.8, T = 3

In Problems 33-38, an object of mass m (in grams) altached to a coiled spring with damping factor b (in grams per second) is pulled down a distance a (in centimeters) from its rest position and then released. Assume that the positive direction of the motion is up and the period is T (in seconds) under simple harmonic motion. (a) Write an equation that relates the distance d of the object from its rest position after t seconds. (b) Graph the equation found in part (a) for 5 oscillations using a graphing utility. m = 10, a = 5, b = 0.7, T = 3

In Problems 39-44, the distance d (in meters) of the bob of a pendulum of mass m (in kilograms) from its rest position at time t (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at t = O? l] (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? d = _20e-0.71/40 cos 5 - 1 600 t

In Problems 39-44, the distance d (in meters) of the bob of a pendulum of mass m (in kilograms) from its rest position at time t (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at t = O? l] (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? d = _20e-0.81/ 4 0 cos 5 - 1600

In Problems 39-44, the distance d (in meters) of the bob of a pendulum of mass m (in kilograms) from its rest position at time t (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at t = O? l] (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? d = -30e-0.61/ 80 cos ( ( 2 7 7 1 ) 2 - 6 0 4 3 0 6 0 t

In Problems 39-44, the distance d (in meters) of the bob of a pendulum of mass m (in kilograms) from its rest position at time t (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at t = O? l] (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? d = -30e- ')/ cos o - /7 0 ( (_71 2 , ) 2 _ 4 0 9 .2 0 5 0 t )

In Problems 39-44, the distance d (in meters) of the bob of a pendulum of mass m (in kilograms) from its rest position at time t (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at t = O? l] (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? ( ( 2 1 ) 3 - 0 . 8 1 - 9 0 0 t )

In Problems 39-44, the distance d (in meters) of the bob of a pendulum of mass m (in kilograms) from its rest position at time t (in seconds) is given. The bob is released from the left of its rest position and represents a negative direction. (a) Describe the motion of the object. Be sure to give the mass and damping factor. (b) What is the initial displacement of the bob? That is, what is the displacement at t = O? l] (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? d = - 10e-0.sl/so cos( ( 271) 2 _ 0.64 t) 3 2500

Loudspeaker A loudspeaker diaphragm is oscillating in simple harmonic motion described by the equation d = a cos(w t) with a frequency of 520 hertz (cycles per second) and a maximum displacement of 0.80 millimeter. Find w and then determine the equation that describes the movement of the diaphragm.

Colossus Added to Six Flags St. Louis in 1986, the Colossus is a giant Ferris wheel. Its diameter is 165 feet, it rotates at a rate of about 1 .6 revolutions per minute, and the bottom of the wheel is 15 feet above the ground. Determine an equation that relates a rider's height above the ground at time t. Assume the passenger begins the ride at the bottom of the wheel. SOl/rce: Six Flags Theme Parks, Inc.

Tuning Fork The end of a tuning fork moves in simple barmonic motion described by tbe equation d = a sin(wt). If a tuning fork for tbe note A above middle C on an eventempered scale (A4' the tone by which an orcbestra tunes itself) bas a frequency of 440 bertz (cycles per second), find w. If the maximum displacement of the end of the tuning fork is 0.01 millimeter, determine the equation that describes the movement of the tuning fork. Source: David Lapp. Physics of Music and Musical Instruments. Medford, MA: Tufts U niversity, 2003

Tuning Fork The end of a tuning fork moves in simple harmonic motion described by the equation d = a sin(wt). If a tuning fork for the note E above middle C on an eventempered scale (E4 ) has a frequency of approximately 329.63 hertz (cycles per second), find w. If the maximum displacement of the end of the tuning fork is 0.025 millimeter, determine the equation that describes the movement of the tuning fork. Source: David Lapp. Physics of Music and Mu.sical Instruments. Medford, MA: Tufts University, 2003

Charging a Capacitor See the illustration. If a charged capacitor is connected to a coil by closing a switch, energy is transferred to the coil and then back to the capacitor in an oscillatory motion. The voltage V (in volts) across the capacitor will gradually diminish to 0 with time t (in seconds). (a) Graph the function relating V and t: Vet) = e-I/3 COS(7Tt), O :s:: t :s:: 3 (b) At what times t will the graph of V touch the graph of y = e-I/3 ? When does the graph of V touch the graph of y = -e-I/3? (c) When will the voltage V be between -0.4 and 0.4 volt? Switch + Capacitor Coil

The Sawtooth Curve An oscilloscope often displays a . sawtooth curve. This curve can be approximated by sinusoidal curves of varying periods and amplitudes. (a) Use a graphing utility to graph the following function, which can be used to approximate the sawtooth curve. f(x) = sin(27Tx) + sin(47Tx), O :s:: x :s:: 4 (b) A better approximation to the sawtooth curve is given by f(x) = sin(27Tx) + sin(47Tx) + sin(87Tx) Use a graphing utility to graph this function for o :s:: x :s:: 4 and compare the result to the graph obtained in part (a). (c) A third and even better approximation to the sawtooth curve is given by f(x) = sin(27Tx) + sin(47Tx) + sin(87Tx) + 6 sin(167Tx) Use a graphing utility to graph this function for o :s:: x :s:: 4 and compare the result to the graphs obtained in parts (a) and (b). (d) What do you think the next approximation to the sawtooth curve is?

Touch-Tone Phones On a Touch-Tone phone, each button produces a unique sound. The sound produced is the sum of two tones, given by y = sin(27Tlt) and y = sin(27Tht) where I and h are the low and high frequencies (cycles per second) shown in the illustration. For example, if you touch 7, the low frequency is I = 852 cycles per second and the high frequency is h = 1209 cycles per second. The sound emitted by touching 7 is y = sin[27T(852)t) + sin[27T(1209)t) Use a graphing utility to graph the sound emitted by touching 7. Touch-Tone phone 1 209 cycles/sec 1477 cycles/sec 1336 cycles/sec

Use a graphing utility to graph the sound emitted by the ,;, key on a Touch-Tone phone. See Problem 5l.

CBL Experiment Pendulum motion is analyzed to esti,. mate simple harmonic motion. A plot is generated with the position of the pendulum over time. The graph is used to find a sinusoidal curve of the form y = A cos [B(x - C) ) + D. Determine the amplitude, period, and frequency. (Activity 16, Real-World Math with the CBL System.)

CBL Experiment The sound from a tuning fork is collected over time. Determine the amplitude, frequency, and period of the graph. A model of the form y = A cos [B( x - C)) is fitted to the data. (Activity 23, Real-World Math with the CBL System.)

Use a . graphing utility to graph the function sm x f(x) = --, x > O. Based on the graph, what do you conx sin x jecture about the value of -- for x close to O? x

Use a graphing utility to graph y = x sin x, y = x2 sin x, and r. y = x 3 sin x for x > O. What patterns do you observe?

U h' t h I . 1 . :J7. se a grap mg utI Ity to grap y = - sm x, y = -2 sm x, an d x x y = sin x for x > O. What patterns do you observe? x

How would you explain to a friend what simple harmonic motion is? How would you explain damped motion?

In Problems 1-4, solve each triangle. 10 A a

In Problems 1-4, solve each triangle. n o b c 35 B 5

In Problems 1-4, solve each triangle. 2 a

In Problems 1-4, solve each triangle. 1 3

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." A = 50, B = 30, a = 1

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." A = 10, C = 40, c = 2

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." A = 100, a = 5, c = 2

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 2, c = 5, A = 60

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 3, c = 1, C= 11 0

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 3, c = 1, C = 20

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 3, c = 1, B= 100

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 3, b = 5, B = 80

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 2, b = 3, c = 1

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 10, b = 7, c = 8

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 1, b = 3, C = 40

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 4, b = 1, C= 100

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 5, b = 3, A = 80

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 2, b = 3, A = 20

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 1, 1 4 b = "2 ' c=-3

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 3, b = 2, c = 2

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 3, A = 10, b = 4

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 4, A = 20, B = 100

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." c = 5, b = 4, A = 70

In Problems 5-24, find the remaining angle(s) and side(s) of each triangle, if it (they) exists. If no triangle exists, say "No triangle." a = 1, b = 2, C = 60

In Problems 25-34, find the area of each triangle. a = 2, b = 3, C = 40

In Problems 25-34, find the area of each triangle. b = 5, c = 5, A = 20

In Problems 25-34, find the area of each triangle. b = 4, C = 10, A = 70

In Problems 25-34, find the area of each triangle. a = 2, b = 1, C = 100

In Problems 25-34, find the area of each triangle. a = 4, b = 3, c = 5

In Problems 25-34, find the area of each triangle. a = 10, b = 7, C = 8

In Problems 25-34, find the area of each triangle. a = 4, b = 2, C = 5

In Problems 25-34, find the area of each triangle. a = 3, b = 2, C = 2

In Problems 25-34, find the area of each triangle. A = 50, B = 30, a = 1

In Problems 25-34, find the area of each triangle. A = 10, C = 40, c = 3

Finding the Grade of a Mountain Trail A straight trail with a uniform inclination leads from a hotel, elevation 5000 feet, to a lake in a valley, elevation 4100 feet. The length of the trail is 4100 feet. What is the inclination (grade) of the trail?

Geometry The hypotenuse of a right triangle is 12 feet. If one leg is 8 feet, find the degree measure of each angle.

Finding the Height of a Helicopter Two observers simultaneously measure the angle of elevation of a helicopter. One angle is measured as 25, the other as 40 (see the figure). If the observers are 1 00 feet apart and the helicopter lies over the line joining them, how high is the helicopter? , , , , , , , , , " ' > t

D etermining Distances at Sea Rebecca, the navigator of a ship at sea, spots two lighthouses that she knows to be 2 miles apart along a straight shoreline. She determines that the angles formed between two line-of-sight observations of the lighthouses and the line from the ship directly to shore are 12 and 30. See the illustration. (a) How far is the ship from lighthouse Ll? (b) How far is the ship from lighthouse L2? ( c) How far is the ship from shore?

Constructing a Highway A highway whose primary directions are north-south is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers 32. a = 3, b = 2, C = 2 decide to go around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of highway needed to go around the bay?

Correcting a Navigational Error A sailboat leaves St. Thomas bound for an island in the British West Indies, 200 miles away. M aintaining a constant speed of 18 miles per hour, but encountering heavy crosswinds and strong currents, the crew finds after 4 hours that the sailboat is off course by 15. (a) How far is the sailboat from the island at this time? (b) Through what angle should the sailboat turn to correct its course? (c) How much time has been added to the trip because of this? (Assume that the speed remains at 18 miles per hour.)

Surveying Two homes are located on opposite sides of a small hill. See the illustration. To measure the distance between them, a surveyor walks a distance of 50 feet from house P to point R, uses a transit to measure L. PRQ, which is found to be 80, and then walks to house Q, a distance of 60 feet. How far apart are the houses? R

Approximating the Area of a Lake To approximate the area of a lake, Cindy walks around the perimeter of the lake, taking the measurements shown in the illustration. Using this technique, what is the approximate area of the lake? [Hint: Use the Law of Cosines on the three triangles shown and then find the sum of their areas.]

Calculating the Cost of Land The irregular parcel of land shown in the figure is being sold for $ 100 per square foot. What is the cost of this parcel? 50 40 1 00 ft

Area of a Segment Find the area of the segment of a circle whose radius is 6 inches formed by a central angle of 50.

Finding the Bearing of a Ship The Majesty leaves the Port at Boston for Bermuda with a bearing of S80E at an average speed of 10 knots. After 1 hour, the ship turns 90 toward the southwest. After 2 hours at an average speed of 20 knots, what is the bearing of the ship from Boston?

Drive Wheels of an Engine TIle drive wheel of an engine is 13 inches in diameter, and the pulley on the rotary pump is 5 inches in diameter. If the shafts of the drive wheel and the pulley are 2 feet apart, what length of belt is required to join them as shown in the figure?

Rework Problem 46 if the belt is crossed, as shown in the figure.

In Problems 48 and 49, an object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period T, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up. a = 3; T = 4 seconds

In Problems 48 and 49, an object attached to a coiled spring is pulled down a distance a from its rest position and then released. Assuming that the motion is simple harmonic with period T, write an equation that relates the displacement d of the object from its rest position after t seconds. Also assume that the positive direction of the motion is up. a = 5; T = 6 seconds

In Problems 50-53, the distance d (in feet) that an object travels in time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is Ihe time required for one oscillation ? (d) What is the frequency? d = 6 sin(2t)

In Problems 50-53, the distance d (in feet) that an object travels in time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is Ihe time required for one oscillation ? (d) What is the frequency? d = 2 cos(4t)

In Problems 50-53, the distance d (in feet) that an object travels in time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is Ihe time required for one oscillation ? (d) What is the frequency? d = -2 cos('1Tt)

In Problems 50-53, the distance d (in feet) that an object travels in time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the maximum displacement from its rest position? (c) What is Ihe time required for one oscillation ? (d) What is the frequency? d = -3 sin[%t]

In Problems 54 and 55, an object of mass m attached to a coiled spring with damping factor b is pulled down a distance a from its rest position and then released. Assume that the positive direction of the motion is up and the period is T under simple harmonic motion. (a) Write an equation that relates the distance d of the object from its rest position after t seconds. (b) Graph Ihe equation found in part (a) for 5 oscillations. m = 40 grams; a = 15 centimeters; b = 0.75 gram/second; T = 5 seconds

In Problems 54 and 55, an object of mass m attached to a coiled spring with damping factor b is pulled down a distance a from its rest position and then released. Assume that the positive direction of the motion is up and the period is T under simple harmonic motion. (a) Write an equation that relates the distance d of the object from its rest position after t seconds. (b) Graph Ihe equation found in part (a) for 5 oscillations. m = 25 grams; a = 13 centimeters; b = 0.65 gram/second; T = 4 seconds

In Problems 56 and 57, the distance d (in meters) of the bob of a pendulum of mass 111. (in kilograms) from its rest position at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the initial displacement of the bob? That is, what is the displacement at t = O? (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? d = _ 1 5e-0 6t/4o cos( C; Y - t)

In Problems 56 and 57, the distance d (in meters) of the bob of a pendulum of mass 111. (in kilograms) from its rest position at time t (in seconds) is given. (a) Describe the motion of the object. (b) What is the initial displacement of the bob? That is, what is the displacement at t = O? (c) Graph the motion using a graphing utility. (d) What is the displacement of the bob at the start of the second oscillation? (e) What happens to the displacement of the bob as time increases without bound? d = _ 20e-0 5t/6o cos( C; Y - t)

In Problems 58 and 59, use the method of adding y-coordinates to graph each function. y = 2 sin x + cos(2x)

In Problems 58 and 59, use the method of adding y-coordinates to graph each function. y = 2 cos(2x) + sm 2