Washington Monument The angle of elevation of the Sun is

tan p 4 + sin 30

Using a calculator, ,rounded to two decimal places. sin 2 =

True or False Exact values can be found for the trigonometric functions of 60

True or False Exact values can be found for the sine of any angle.

Write down the exact value of each of the six trigonometric functions of 45.

Write down the exact value of each of the six trigonometric functions of 30 and of 60.

In Problems 716, f(u) = sin u and g(u) = cos u. Find the exact value of each expression if u = 60. Do not use a calculator.f(u)

In Problems 716, f(u) = sin u and g(u) = cos u. Find the exact value of each expression if u = 60. Do not use a calculator.g(u)

In Problems 716, f(u) = sin u and g(u) = cos u. Find the exact value of each expression if u = 60. Do not use a calculator.f(u2)

In Problems 716, f(u) = sin u and g(u) = cos u. Find the exact value of each expression if u = 60. Do not use a calculator.ga u 2 fa b

In Problems 716, f(u) = sin u and g(u) = cos u. Find the exact value of each expression if u = 60. Do not use a calculator.[f(u2] 2

In Problems 716, f(u) = sin u and g(u) = cos u. Find the exact value of each expression if u = 60. Do not use a calculator.[g(u2

In Problems 716, f(u) = sin u and g(u) = cos u. Find the exact value of each expression if u = 60. Do not use a calculator.2 f(u)

In Problems 716, f(u) = sin u and g(u) = cos u. Find the exact value of each expression if u = 60. Do not use a calculator.2g(u)

In Problems 716, f(u) = sin u and g(u) = cos u. Find the exact value of each expression if u = 60. Do not use a calculator.f(u) 2

In Problems 716, f(u) = sin u and g(u) = cos u. Find the exact value of each expression if u = 60. Do not use a calculator.g(u) 2

In Problems 1728, find the exact value of each expression. Do not use a calculator. 4 cos 45 - 2 sin 45

In Problems 1728, find the exact value of each expression. Do not use a calculator. 2 sin 45 + 4 cos 30

In Problems 1728, find the exact value of each expression. Do not use a calculator. 6 tan 45 - 8 cos 60

In Problems 1728, find the exact value of each expression. Do not use a calculator.sin 30 # tan 60

In Problems 1728, find the exact value of each expression. Do not use a calculator.sec p 4 + 2 csc p 3

In Problems 1728, find the exact value of each expression. Do not use a calculator.tan p 4 + cot p 4

In Problems 1728, find the exact value of each expression. Do not use a calculator.sec2 p 6 - 4

In Problems 1728, find the exact value of each expression. Do not use a calculator.4 + 60 tan2 p 3

In Problems 1728, find the exact value of each expression. Do not use a calculator.in2 30 + cos2 4 + 60

In Problems 1728, find the exact value of each expression. Do not use a calculator.sec 60 2 60 - tan2 45

In Problems 1728, find the exact value of each expression. Do not use a calculator.1 - cos 45 2 30 - cos2 sec 60

In Problems 1728, find the exact value of each expression. Do not use a calculator.1 + tan2 30 - csc2 1 - cos 45

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places. sin 28

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places. cos 14

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places. tan 21

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places. cot 70

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places. sec 41

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places. csc 55

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.sin p 10

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.cos p 8

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.tan 5p 12

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.cot p 18

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.sec p 12

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.csc 5p 13

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.sin 1

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.tan 1

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.sin 1

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.tan 1

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.tan 0.3

In Problems 2946, use a calculator to find the approximate value of each expression. Round the answer to two decimal places.tan 0.1

In Problems 4756, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = . Find the value of each of the following:(f + g)(30)

In Problems 4756, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = . Find the value of each of the following:(f - g)(60)

In Problems 4756, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = . Find the value of each of the following:(f b # g)

In Problems 4756, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = . Find the value of each of the following:(f # g)a p 3 b

In Problems 4756, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = . Find the value of each of the following:(f # h) p6

In Problems 4756, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = . Find the value of each of the following:(g # p)(60)

In Problems 4756, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = . Find the value of each of the following:(p # g)(45)

In Problems 4756, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = . Find the value of each of the following:(h # f) p6

In Problems 4756, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = . Find the value of each of the following:(a) Find .What point is on the graph of f ? (b) Assuming f is one-to-one. Use the result of part (a), to find a point on the graph of . (c) What point is on the graph of y = fax + p 4 b - 3 if x = p 4 ?

In Problems 4756, f(x) = sin x, g(x) = cos x, h(x) = 2x, and p(x) = . Find the value of each of the following:(a) Find .What point is on the graph of g? (b) Assuming g is one-to-one. Use the result of part (a), to find a point on the graph of . (c) What point is on the graph of y = 2gax - p 6 b if x = p 6 ?

Problems 5760 require the following discussion. Projectile Motion The path of a projectile fired at an inclination to the horizontal with initial speed is a parabola. See the figure. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the function where feet per second per second meters per second per second is the acceleration due to gravity. The maximum height H of the projectile is given by theR1u2 = 2v0 2 sin u cos u g H1u2 = v02 sin2 u2g In Problems 5760, find the range R and maximum height H of the projectile. Round answers to two decimal places. The projectile is fired at an angle of 45 to the horizontalwith an initial speed of 100 feet per second.

Problems 5760 require the following discussion. Projectile Motion The path of a projectile fired at an inclination to the horizontal with initial speed is a parabola. See the figure. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the function where feet per second per second meters per second per second is the acceleration due to gravity. The maximum height H of the projectile is given by theR1u2 = 2v0 2 sin u cos u g H1u2 = v02 sin2 u2g In Problems 5760, find the range R and maximum height H of the projectile. Round answers to two decimal places. The projectile is fired at an angle of 30 to the horizontal with an initial speed of 150 meters per second.

Problems 5760 require the following discussion. Projectile Motion The path of a projectile fired at an inclination to the horizontal with initial speed is a parabola. See the figure. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the function where feet per second per second meters per second per second is the acceleration due to gravity. The maximum height H of the projectile is given by theR1u2 = 2v0 2 sin u cos u g H1u2 = v02 sin2 u2g In Problems 5760, find the range R and maximum height H of the projectile. Round answers to two decimal places. The projectile is fired at an angle of 25 to the horizontal with an initial speed of 500 meters per second.

Problems 5760 require the following discussion. Projectile Motion The path of a projectile fired at an inclination to the horizontal with initial speed is a parabola. See the figure. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the function where feet per second per second meters per second per second is the acceleration due to gravity. The maximum height H of the projectile is given by theR1u2 = 2v0 2 sin u cos u g H1u2 = v02 sin2 u2g In Problems 5760, find the range R and maximum height H of the projectile. Round answers to two decimal places. The projectile is fired at an angle of 50 to the horizontal with an initial speed of 200 feet per second.

Inclined Plane See the illustration. If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is modeled by the function where a is the length (in feet) of the base and per second per second is the acceleration due to gravity. How long does it take a block to slide down an inclined plane with base when (a) (b) (c) u = 60? u = 45? u = 30? a = 10 feet g L 32 feet t1u2 = A 2a g sin u cos u

Piston Engines See the illustration. In a certain piston engine,the distance x (in inches) from the center of the driveshaft to the head of the piston is modeled by the function x1u2 = cos u + 416 + 0.512 cos2 u - 12 where is the angle between the crank and the path of the piston head. Find x when and when

Calculating the Time of a Trip Two oceanfront homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved path that parallels the ocean. Sally can jog 8 miles per hour along the paved path, but only 3 miles per hour in the sand on the beach. Because a river flows between the two houses, it is necessary to jog on the sand to the path, continue on the path, and then jog on the sand to get from one house to the other. See the illustration.(a) Express the time T to get from one house to the other as a function of the angle shown in the illustration. (b) Calculate the time T for . How long is Sally on the paved path? (c) Calculate the time T for . How long is Sally on the paved path? (d) Calculate the time T for . How long is Sally on the paved path? (e) Calculate the time T for . Describe the route taken. (f) Calculate the time T for Describe the route taken. Explain why must be larger than 14. (g) Graph .What angle results in the least time? What is the least time? How long is Sally on the paved path?

Designing Fine Decorative Pieces A designer of decorative art plans to market solid gold spheres encased in clear crystal cones. Each sphere is of fixed radius R and will be enclosed in a cone of height h and radius r. See the illustration on page 538. Many cones can be used to enclose the sphere, each having a different slant angle u.(a) Express the volume V of the cone as a function of the slant angle of the cone. [Hint: The volume V of a cone of height h and radius r is .] (b) What volume V is required to enclose a sphere of radius 2 centimeters in a cone whose slant angle is 30? 45? 60? (c) What slant angle should be used for the volume V of the cone to be a minimum? (This choice minimizes the amount of crystal required and gives maximum emphasis to the gold sphere.)

Geometry A right triangle has a hypotenuse of length 8 inches. If one angle is 35, find the length of each leg.

Geometry A right triangle has a hypotenuse of length 10 centimeters.If one angle is 40,find the length of each leg

Geometry A right triangle contains a 25 angle. (a) If one leg is of length 5 inches, what is the length of the hypotenuse? (b) There are two answers. How is this possible?

Geometry A right triangle contains an angle of radian. (a) If one leg is of length 3 meters, what is the length of the hypotenuse? (b) There are two answers. How is this possible?

Finding the Width of a Gorge Find the distance from A to C across the gorge illustrated in the figure.

Finding the Distance across a Pond Find the distance from A to C across the pond illustrated in the figure.

The Eiffel Tower The tallest tower built before the era of television masts,the EiffelTower was completed on March 31, 1889. Find the height of the Eiffel Tower (before a television mast was added to the top) using the information given in the illustration.

Finding the Distance of a Ship from Shore A person in a small boat, offshore from a vertical cliff known to be 100 feet in height,takes a sighting of the top of the cliff.If the angle of elevation is found to be 30, how far offshore is the boat?

Finding the Distance to a Plateau Suppose that you are headed toward a plateau 50 meters high. If the angle of elevation to the top of the plateau is 60, how far are you from the base of the plateau?

Finding the Reach of a Ladder A 22-foot extension ladder leaning against a building makes a 70 angle with the ground. How far up the building does the ladder touch?

Finding the Distance between Two Objects A blimp flying at an altitude of 500 feet, lies directly over a line from Soldier Field to the Adler Planetarium on Lake Michigan (see the figure). If the angle of depression from the blimp to the stadium is 32 and from the blimp to the planetarium is 23, find the distance between Soldier Field and the Adler Planetarium.

Hot-air Balloon While taking a ride in a hot-air balloon in Napa Valley, Francisco wonders how high he is. To find out, he chooses a landmark that is to the east of the balloon and measures the angle of depression to be 54. A few minutes later, after traveling 100 feet east, the angle of depression to the same landmark is determined to be 61. Use this information to determine the height of the balloon.

Mt. Rushmore To measure the height of Lincolns caricature on Mt. Rushmore, two sightings 800 feet from the base of the mountain are taken. If the angle of elevation to the bottom of Lincolns face is 32 and the angle of elevation to the top is 35, what is the height of Lincolns face?

The CN Tower The CN Tower, located in Toronto, Canada, is the tallest structure in the Americas.While visitingToronto, a tourist wondered what the height of the tower above the top of the Sky Pod is. While standing 4000 feet from the tower, she measured the angle to the top of the Sky Pod to be 20.1. At this same distance, the angle of elevation to the top of the tower was found to be 24.4. Use this information to determine the height of the tower above the Sky Pod

Finding the Length of a Guy Wire A radio transmission tower is 200 feet high. How long should a guy wire be if it is to be attached to the tower 10 feet from the top and is to make an angle of 45 with the ground?

Finding the Height of a Tower A guy wire 80 feet long is attached to the top of a radio transmission tower, making an angle of 45 with the ground. How high is the tower?

Washington Monument The angle of elevation of the Sun is 35.1 at the instant the shadow cast by the Washington Monument is 789 feet long. Use this information to calculate the height of the monument.

Finding the Length of a MountainTrail A straight trail with an inclination of 17 leads from a hotel at an elevation of 9000 feet to a mountain lake at an elevation of 11,200 feet. What is the length of the trail?

Constructing a Highway A highway whose primary directions are northsouth 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?

Photography A camera is mounted on a tripod 4 feet high at a distance of 10 feet from George, who is 6 feet tall. See the illustration. If the camera lens has angles of depression and elevation of 20, will Georges feet and head be seen by the lens? If not, how far back will the camera need to be moved to include Georges feet and head?

Calculating Pool Shots A pool player located at X wants to shoot the white ball off the top cushion and hit the red ball dead center. He knows from physics that the white ball will come off a cushion at the same angle as it hits a cushion.Where on the top cushion should he hit the white ball?

The Freedom Tower The Freedom Tower is to be the centerpiece of the rebuilding of the World Trade Center in NewYork City.The tower will be 1776 feet tall (not including a broadcast antenna).The angle of elevation from the base of an office building to the top of the tower is 34.The angle of elevation from the helipad on the roof of the office building to the top of the tower is 20.(a) How far away is the office building from the Freedom Tower? Assume the side of the tower is vertical. Round to the nearest foot. (b) How tall is the office building? Round to the nearest foot.

Use a calculator set in radian mode to complete the following table. What can you conclude about the value of as u approaches 0?

Use a calculator set in radian mode to complete the following table.What can you conclude about the value of as u approaches 0?

Find the exact value oftan 1 # tan 2 # tan 3 # # tan 89

Find the exact value ofcot 1 # cot 2 # cot 3 # # cot 89

Find the exact value ofcos 1 # cos 2 # # cos 45 # csc 46 # # csc 89

Find the exact value ofsin 1 # sin 2 # # sin 45 # sec 46 # # sec 89

Guy Wires Two poles, 10 feet and 15 feet in height, are located 30 feet apart. They are to be supported by a single guy wire attached to their tops and anchored to the ground at a point on the line joining the two poles. See the illustration. Suppose is the angle between the guy wire and the taller pole and L is the length of the guy wire. (a) Express as a function of u. (b) What is the domain of L? (c) Find L if . What is the corresponding value of x? (d) Graph L and find the angle that minimizes L. (e) What is the minimum length for the guy wire? Where is it anchored to the ground?

Write a brief paragraph that explains how to quickly compute the trigonometric functions of30 45 60

Explain how you would measure the width of the Grand Canyon from a point on its ridge.

Explain how you would measure the height of a TV tower that is on the roof of a tall building