Approaching a lighthouse A boat approaches a 50-ft-high

Define the six trigonometric functions in terms of the sides of a right triangle.

Explain how a point P1x, y2 on a circle of radius r determines an angle u and the values of the six trigonometric functions at u.

How is the radian measure of an angle determined?

Explain what is meant by the period of a trigonometric function. What are the periods of the six trigonometric functions?

What are the three Pythagorean identities for the trigonometric functions?

How are the sine and cosine functions related to the other four trigonometric functions?

Where is the tangent function undefined?

What is the domain of the secant function?

Explain why the domain of the sine function must be restricted in order to define its inverse function.

Why do the values of cos-1 x lie in the interval 30, p4?

Is it true that tan 1tan-1 x2 = x for all x? Is it true that tan-1 1tan x2 = x for all x?

Sketch the graphs of y = cos x and y = cos-1 x on the same set of axes.

The function tan x is undefined at x = {p>2. How does this fact appear in the graph of y = tan-1 x?

State the domain and range of sec-1 x. .

1522. Evaluating trigonometric functions Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians cos 12p>32

1522. Evaluating trigonometric functions Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians sin 12p>32

1522. Evaluating trigonometric functions Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians tan 1-3p>42

1522. Evaluating trigonometric functions Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians tan 115p>42

1522. Evaluating trigonometric functions Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians cot 1-13p>32

1522. Evaluating trigonometric functions Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians sec 17p>62

1522. Evaluating trigonometric functions Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians cot 1-17p>32

1522. Evaluating trigonometric functions Evaluate the following expressions using a unit circle. Use a calculator to check your work. All angles are in radians sin 116p>32

2328. Evaluating trigonometric functions Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work. cos 0

2328. Evaluating trigonometric functions Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work. sin 1-p>22

2328. Evaluating trigonometric functions Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work. cos 1-p2

2328. Evaluating trigonometric functions Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work. tan 3p

2328. Evaluating trigonometric functions Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work. sec 15p>22

2328. Evaluating trigonometric functions Evaluate the following expressions or state that the quantity is undefined. Use a calculator to check your work. cot p

2936. Trigonometric identities Prove that sec u = 1 cos u .

2936. Trigonometric identities Prove that tan u = sin u cos u .

2936. Trigonometric identities Prove that tan2 u + 1 = sec2 u.

2936. Trigonometric identities Prove that sin u csc u + cos u sec u = 1.

2936. Trigonometric identities Prove that sec 1p>2 - u2 = csc u.

2936. Trigonometric identities Prove that sec 1x + p2 = -sec x.

2936. Trigonometric identities Find the exact value of cos 1p>122.

2936. Trigonometric identities Find the exact value of tan 13p>82.

3746. Solving trigonometric equations Solve the following equations. tan x = 1

3746. Solving trigonometric equations Solve the following equations. 2u cos u + u = 0

3746. Solving trigonometric equations Solve the following equations. sin2 u = 14 , 0 u 6 2p

3746. Solving trigonometric equations Solve the following equations. cos2 u = 12 , 0 u 6 2p

3746. Solving trigonometric equations Solve the following equations. 12 sin x - 1 = 0

3746. Solving trigonometric equations Solve the following equations. sin 3x = 22 2 , 0 x 6 2p

3746. Solving trigonometric equations Solve the following equations. cos 3x = sin 3x, 0 x 6 2p

3746. Solving trigonometric equations Solve the following equations. sin2 u - 1 = 0

3746. Solving trigonometric equations Solve the following equations. sin u cos u = 0, 0 u 6 2p

3746. Solving trigonometric equations Solve the following equations. tan2 2u = 1, 0 u 6 p

4756. Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. sin-1 1

4756. Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. cos-1 1-12

4756. Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. tan-1 1

4756. Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. cos-1 1 - 22 2 2

4756. Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. sin-1 23 2

4756. Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. cos-1 2

4756. Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. cos-1 1 -12 2

4756. Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. sin-1 1-12

4756. Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. cos 1cos-1 1-122

4756. Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. cos-1 1cos 17p>622

5762. Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 7 0. cos 1sin-1 x2

5762. Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 7 0. cos 1sin-1 1x>322

5762. Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 7 0. sin 1cos-1 1x>222

5762. Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 7 0. sin-1 1cos u2, for 0 u p>2

5762. Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 7 0. sin 12 cos-1 x2 1Hint: Use sin 2u = 2 sin u cos u.2

5762. Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 7 0. cos 12 sin-1 x2 1Hint: Use cos 2u = cos2 u - sin2 u.2

6364. Identities Prove the following identities. cos-1 x + cos-1 1-x2 = p

6364. Identities Prove the following identities. sin-1 y + sin-1 1-y2 = 0

6566. Verifying identities Sketch a graph of the given pair of functions to conjecture a relationship between the two functions. Then verify the conjecture. sin-1 x; p 2 - cos-1 x

6566. Verifying identities Sketch a graph of the given pair of functions to conjecture a relationship between the two functions. Then verify the conjecture. tan-1 x; p 2 - cot-1 x

6774. Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. tan-1 13

6774. Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. cot-1 1-1>132

6774. Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. sec-1 2

6774. Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. csc-1 1-12

6774. Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. tan-1 1tan 1p>422

6774. Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. tan-1 1tan 13p>422

6774. Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. csc-1 1sec 22

6774. Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. tan 1tan-1 12

7580. Right-triangle relationships Use a right triangle to simplify the given expressions. Assume x 7 0. cos 1tan-1 x2

7580. Right-triangle relationships Use a right triangle to simplify the given expressions. Assume x 7 0. tan 1cos-1 x2

7580. Right-triangle relationships Use a right triangle to simplify the given expressions. Assume x 7 0. cos 1sec-1 x2

7580. Right-triangle relationships Use a right triangle to simplify the given expressions. Assume x 7 0. cot 1tan-1 2x2

7580. Right-triangle relationships Use a right triangle to simplify the given expressions. Assume x 7 0. sin asec-1 a 2x2 + 16 4 b b

7580. Right-triangle relationships Use a right triangle to simplify the given expressions. Assume x 7 0. cos atan-1 a x 29 - x2 b b .

8182. Right-triangle pictures Express u in terms of x using the inverse sine, inverse tangent, and inverse secant functions.

8182. Right-triangle pictures Express u in terms of x using the inverse sine, inverse tangent, and inverse secant functions.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. sin 1a + b2 = sin a + sin b. b. The equation cos u = 2 has multiple real solutions. c. The equation sin u = 12 has exactly one solution. d. The function sin 1px>122 has a period of 12. e. Of the six basic trigonometric functions, only tangent and cotangent have a range of 1- _, _2. f. sin-1 x cos-1 x = tan-1 x. g. cos-1 1cos 115p>1622 = 15p>16. h. sin-1 x = 1>sin x.

8487. One function gives all six Given the following information about one trigonometric function, evaluate the other five functions. sin u = - 4 5 and p 6 u 6 3p>2

8487. One function gives all six Given the following information about one trigonometric function, evaluate the other five functions. cos u = 5 13 and 0 6 u 6 p>2

8487. One function gives all six Given the following information about one trigonometric function, evaluate the other five functions. sec u = 5 3 and 3p>2 6 u 6 2p

8487. One function gives all six Given the following information about one trigonometric function, evaluate the other five functions. csc u = 13 12 and 0 6 u 6 p>2

8891. Amplitude and period Identify the amplitude and period of the following functions. f 1u2 = 2 sin 2u

8891. Amplitude and period Identify the amplitude and period of the following functions. g1u2 = 3 cos 1u>32

8891. Amplitude and period Identify the amplitude and period of the following functions. p1t2 = 2.5 sin 112 1t - 322

8891. Amplitude and period Identify the amplitude and period of the following functions. q1x2 = 3.6 cos 1px>242

9295. Graphing sine and cosine functions Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. f 1x2 = 3 sin 2x

9295. Graphing sine and cosine functions Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. g1x2 = -2 cos 1x>32

9295. Graphing sine and cosine functions Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. p1x2 = 3 sin 12x - p>32 + 1

9295. Graphing sine and cosine functions Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. q1x2 = 3.6 cos 1px>242 + 2

9697. Designer functions Design a sine function with the given properties. It has a period of 12 hr with a minimum value of -4 at t = 0 hr and a maximum value of 4 at t = 6 hr.

9697. Designer functions Design a sine function with the given properties. It has a period of 24 hr with a minimum value of 10 at t = 3 hr and a maximum value of 16 at t = 15 hr.

Field goal attempt Near the end of the 1950 Rose Bowl football game between the University of California and Ohio State University, Ohio State was preparing to attempt a field goal from a distance of 23 yd from the endline at point A on the edge of the kicking region (see figure). But before the kick, Ohio State committed a penalty and the ball was backed up 5 yd to point B on the edge of the kicking region. After the game, the Ohio State coach claimed that his team deliberately committed a penalty to improve the kicking angle. Given that a successful kick must go between the uprights of the goal posts G1 and G2, is _G1BG2 greater than _G1AG2? (In 1950, the uprights were 23 ft 4 in apart, equidistant from the origin on the end line. The boundaries of the kicking region are 53 ft 4 in apart and are equidistant from the y-axis. (Source: The College Mathematics Journal 27, 4, Sep 1996)

A surprising result The Earth is approximately circular in cross section, with a circumference at the equator of 24,882 miles. Suppose we use two ropes to create two concentric circles: one by wrapping a rope around the equator and another using a rope 38 ft longer (see figure). How much space is between the ropes?

Daylight function for 40_ N Verify that the function has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset. a. It has a period of 365 days. b. Its maximum and minimum values are 14.8 and 9.2, respectively, which occur approximately at t = 172 and t = 355, respectively (corresponding to the solstices). c. D1812 = 12 and D12642 _ 12 (corresponding to the equinoxes).

Block on a spring A light block hangs at rest from the end of a spring when it is pulled down 10 cm and released. Assume the block oscillates with an amplitude of 10 cm on either side of its rest position with a period of 1.5 s. Find a trigonometric function

Approaching a lighthouse A boat approaches a 50-ft-high lighthouse whose base is at sea level. Let d be the distance between the boat and the base of the lighthouse. Let L be the distance between the boat and the top of the lighthouse. Let u be the angle of elevation between the boat and the top of the lighthouse. a. Express d as a function of u. b. Express L as a function of u.

Ladders Two ladders of length a lean against opposite walls of an alley with their feet touching (see figure). One ladder extends h feet up the wall and makes a 75 angle with the ground. The other ladder extends k feet up the opposite wall and makes a 45 angle with the ground. Find the width of the alley in terms of h. Assume the ground is horizontal and perpendicular to both walls. _

Pole in a corner A pole of length L is carried horizontally around a corner where a 3-ft-wide hallway meets a 4-ft-wide hallway. For 0 6 u 6 p>2, find the relationship between L and u at the moment when the pole simultaneously touches both walls and the corner P. Estimate u when L = 10 ft.

Little-known fact The shortest day of the year occurs on the winter solstice (near December 21) and the longest day of the year occurs on the summer solstice (near June 21). However, the latest sunrise and the earliest sunset do not occur on the winter solstice, and the earliest sunrise and the latest sunset do not occur on the summer solstice. At latitude 40 north, the latest sunrise occurs on January 4 at 7:25 a.m. (14 days after the solstice), and the earliest sunset occurs on December 7 at 4:37 p.m. (14 days before the solstice). Similarly, the earliest sunrise occurs on July 2 at 4:30 a.m. (14 days after the solstice) and the latest sunset occurs on June 7 at 7:32 p.m. (14 days before the solstice). Using sine functions, devise a function s1t2 that gives the time of sunrise t days after January 1 and a function S1t2 that gives the time of sunset t days after January 1. Assume that s and S are measured in minutes and s = 0 and S = 0 correspond to 4:00 a.m. Graph the functions. Then graph the length of the day function D1t2 = S1t2 - s1t2 and show that the longest and shortest days occur on the solstices.

Viewing angles An auditorium with a flat floor has a large flatpanel television on one wall. The lower edge of the television is 3 ft above the floor, and the upper edge is 10 ft above the floor (see figure). Express u in terms of x.

Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle u (measured in radians) is A = 12 r2 u. u r

Law of cosines Use the figure to prove the law of cosines (which is a generalization of the Pythagorean theorem): c2 = a2 + b2 - 2ab cos u.

Law of sines Use the figure to prove the law of sines: sin A a = sin B b = sin C