MORE TRIGONOMETRIC FUNCTIONS

Transcription

1 CHAPTER MORE TRIGONOMETRIC FUNCTIONS The relationshis among the lengths of the sides of an isosceles right triangle or of the right triangles formed by the altitude to a side of an equilateral triangle make it ossible for us to find exact values of the trig functions of angles of 30, 45, and their multiles. How are the trigonometric function values for other angles determined? Before calculators and comuters were readily available, handbooks that contained tables of values were a common tool of mathematicians. Now calculators or comuters will return these values. How were the values in these tables or those dislayed by a calculator determined? In more advanced math courses, you will learn that trigonometric function values can be aroximated by evaluating an infinite series to the required number of decimal laces. x sin x 5 x 2 3 3! 5! x5 2 7! x7 c To use this formula correctly, x must be exressed in a unit of measure that has no dimension.this unit of measure, which we will define in this chater, is called a radian. 0 CHAPTER TABLE OF CONTENTS 0- Radian Measure 0-2 Trigonometric Function Values and Radian Measure 0-3 Pythagorean Identities 0-4 Domain and Range of Trigonometric Functions 0-5 Inverse Trigonometric Functions 0-6 Cofunctions Chater Summary Vocabulary Review Exercises Cumulative Review 399

2 400 More Trigonometric Functions 0- RADIAN MEASURE A line segment can be measured in different units of measure such as feet, inches, centimeters, and meters. Angles and arcs can also be measured in different units of measure. We have measured angles in degrees, but another frequently used unit of measure for angles is called a radian. DEFINITION A radian is the unit of measure of a central angle that intercets an arc equal in length to the radius of the circle. cm cm 3 cm cm cm In a circle of radius centimeter, a central angle whose measure is radian intercets an arc whose length is centimeter. The radian measure of any central angle in a circle of radius is equal to the length of the arc that it intercets. In a circle of radius centimeter, a central angle of 2 radians intercets an arc whose length is 2 centimeters, and a central angle of 3 radians intercets an arc whose length is 3 centimeters. In a circle of radius 2 centimeters, a central angle whose measure is radian intercets an arc whose length is 2 centimeters, a central angle of 2 radians intercets an arc whose measure is 4 centimeters, and a central angle of 3 radians intercets an arc whose measure is 6 centimeters. In general, the radian measure u of a central angle of a circle is the length of the interceted arc, s, divided by the radius of the circle, r. u 5 s r 2 cm 2 cm 6 cm cm 2 cm 2 cm cm Relationshi Between Degrees and Radians When the diameter of a circle is drawn, the straight angle whose vertex is at the center of the circle intercets an arc that is a semicircle. The length of the semicircle is one-half the circumference of the circle or 2 (2r) 5r. Therefore, the radian measure of a r straight angle is r 5. Since the degree measure of a straight angle is 80, radians and 80 degrees are measures of the same angle.we can write the following relationshi: radians 5 80 degrees r O r

3 Radian Measure 40 We will use this relationshi to change the degree measure of any angle to radian measure. Changing Degrees to Radians The measure of any angle can be exressed in degrees or in radians. We can do this by forming the following roortion: measure in degrees measure in radians 5 80 This roortion can be used to find the radian measure of an angle whose degree measure is known. For examle, to find u, the radian measure of an angle of 30, substitute 30 for measure in degrees and u for measure in radians in the roortion. Then use the fact that the roduct of the means is equal to the roduct of the extremes to solve for u. 30 u u 5 30 u 5 30 u The radian measure of an angle of 30 is 6. We know that sin Use a calculator to comare this value to sin u when u is 6 radians. Begin by changing the calculator to radian mode. Then enter the exression. ENTER: MODE ENTER: SIN 2nd 6 ) ENTER MATH ENTER ENTER DISPLAY: Normal Float Radian Sci Eng Degree DISPLAY: sin(π /6) Frac /2 The calculator confirms that sin 30 5 sin 6.

4 402 More Trigonometric Functions Changing Radians to Degrees When the radian measure of an angle is known, we can use either of the two methods, shown below, to find the degree measure of the angle. METHOD Proortion Use the same roortion that was used to change from degrees to radians. For examle, to convert an angle of 2 radians to degrees, let u5degree measure of the angle. () Write the roortion that relates degree measure to radian measure: measure in degrees measure in radians 5 80 (2) Substitute u for the degree measure and 2 for the radian measure: u A 2 B 5 80 (3) Solve for u: The measure of the angle is 90. u 5 2 (80) u 5 90 u590 METHOD 2 Substitution Since we know that radians 5 80, substitute 80 for radians and simlify the exression. For examle: 2 radians 5 2 ( radians) 5 2(80 ) 5 90 Radian measure is a ratio of the length of an arc to the length of the radius of a circle.these ratios must be exressed in the same unit of measure.therefore, the units cancel and the ratio is a number that has no unit of measure. EXAMPLE The radius of a circle is 4 centimeters. A central angle of the circle intercets an arc of 2 centimeters. a. What is the radian measure of the angle? b. Is the angle acute, obtuse, or larger than a straight angle?

5 Radian Measure 403 Solution a. How to Proceed () Write the rule that states that the radian measure of a central angle, u, is equal to the length s of the interceted arc divided by the length r of a radius: (2) Substitute the given values: (3) Simlify the fraction: u5 s r u5 2 4 u b. Since we know that radians 5 80, then radian 5. 3 radians 5 3 A Therefore, the angle is obtuse. Answers a. 3 radians b. Obtuse B EXAMPLE 2 Solution The radius of a circle is 8 inches. What is the length of the arc interceted by a central angle of the circle if the measure of the angle is 2.5 radians? u5 s r 8 in s s 5 2.5(8) 5 20 Answer 20 inches EXAMPLE 3 What is the radian measure of an angle of 45? Solution How to Proceed () Write the roortion: (2) Substitute x for the radian measure and 45 for the degree measure: (3) Solve for x: measure in degrees measure in radians x x 5 45 x (4) Exress x in simlest form: x 5 4 Answer or 4 4

6 404 More Trigonometric Functions EXAMPLE 4 5 The radian measure of an angle is 6. a. What is the measure of the angle in degrees? b. What is the radian measure of the reference angle? 5 6 Solution a. METHOD : Proortions METHOD 2: Substitution Let u 5degree measure of the angle. measure in degrees measure in radians 5 80 u u 5 80 A 5 6 B u 5 50 u550 Substitute 80 for radians, and simlify. 5 6 radians ( radians) (808) b. Since 6 is a second-quadrant angle, its reference angle is equal to or, in radians: Answers a. 50 b. 6 Exercises Writing About Mathematics. Ryan and Rebecca each found the radian measure of a central angle by measuring the radius of the circle and the length of the interceted arc. Ryan used inches and Rebecca used centimeters when making their measurements. If Ryan and Rebecca each measured accurately, will the measures that they obtain for the angle be equal? Justify your answer. 2. If a wheel makes two comlete revolutions, each soke on the wheel turns through an angle of how many radians? Exlain your answer. Develoing Skills In 3 2, find the radian measure of each angle whose degree measure is given

7 In 3 22, find the degree measure of each angle whose radian measure is given Radian Measure 405 In 23 27, for each angle with the given radian measure: a. Give the measure of the angle in degrees. b. Give the measure of the reference angle in radians. c. Draw the angle in standard osition and its reference angle as an acute angle formed by the terminal side of the angle and the x-axis In 28 37, u is the radian measure of a central angle that intercets an arc of length s in a circle with a radius of length r. 28. If s 5 6 and r 5, find u. 29. If u54.5 and s 5 9, find r. 30. If u52.5 and r 5 0, find s. 3. If r 5 2 and u5.6, find s. 32. If r and s 5 5, find u. 33. If s 5 6 and u50.4, find r If r and s 5 2, find u. 35. If r 5 6 and u5 3, find s. 36. If s 5 8 and u5 6 5, find r. 37. If u56 and r 5, find s. 38. Circle O has a radius of.7 inches. What is the length, in inches, of an arc interceted by a central angle whose measure is 2 radians? 39. In a circle whose radius measures 5 feet, a central angle intercets an arc of length 2 feet. Find the radian measure of the central angle. 40. The central angle of circle O has a measure of 4.2 radians and it intercets an arc whose length is 6.3 meters. What is the length, in meters, of the radius of the circle? 4. Comlete the following table, exressing degree measures in radian measure in terms of. Degrees Radians Alying Skills 42. The endulum of a clock makes an angle of 2.5 radians as its ti travels 8 feet. What is the length of the endulum? 43. A wheel whose radius measures 6 inches is rotated. If a oint on the circumference of the wheel moves through an arc of 2 feet, what is the measure, in radians, of the angle through which a soke of the wheel travels?

8 406 More Trigonometric Functions 44. The wheels on a bicycle have a radius of 40 centimeters. The wheels on a cart have a radius of 0 centimeters. The wheels of the bicycle and the wheels of the cart all make one comlete revolution. a. Do the wheels of the bicycle rotate through the same angle as the wheels of the cart? Justify your answer. b. Does the bicycle travel the same distance as the cart? Justify your answer. 45. Latitude reresents the measure of a central angle with vertex at the center of the earth, its initial side assing through a oint on the equator, and its terminal side assing through the given location. (See the figure.) Cities A and B are on a north-south line. City A is located at 30 N and City B is located at 52 N. If the radius of the earth is aroximately 6,400 kilometers, find d, the distance between the two cities along the circumference of the earth. Assume that the earth is a erfect shere. North Pole City B City A N 6,400 km Equator 0-2 TRIGONOMETRIC FUNCTION VALUES AND RADIAN MEASURE Using Radians to Find Trigonometric Function Values Hands-On Activity In revious chaters, we found the exact trigonometric function values of angles of 0 o,30 o,45 o,60 o,90 o, 80, and 270. Coy and comlete the table below to show measures of these angles in radians and the corresonding function values. Let u be the measure of an angle in standard osition. u in Degrees u in Radians sin u cos u tan u We can use these function values to find the exact function value of any angle whose radian measure is a multile of 6 or 4. EXAMPLE Find the exact value of tan 2 3.

9 Trigonometric Function Values and Radian Measure 407 Solution METHOD METHOD 2 y 2 3 (60 ) 3 (20 ) x Answer 2 The radian measure 3 is equivalent 2(808) to the degree measure An angle of 20 is a second-quadrant angle whose tangent is negative. The reference angle is tan tan 20 52tan !3 2!3 2 Since 2, 2 3,, an angle of 3 is a second-quadrant angle. The reference angle of an angle 2 of radians is The tangent of a second-quadrant angle is negative. 2 tan 3 52tan 3 52!3 The grahing calculator will return the trigonometric function values of angles entered in degree or radian measure. Before entering an angle measure, be sure that the calculator is in the correct mode. For examle, to find tan 8 to four decimal laces, use the following sequence of keys and round the number given by the calculator. Be sure that your calculator is set to radian mode. ENTER: TAN 2nd 8 DISPLAY: tan(π /8) ) ENTER Therefore, tan EXAMPLE 2 Find the value of cos u to four decimal laces when u radians. Solution ENTER: COS 2.75 ) ENTER DISPLAY: Note that the cosine is negative because the angle is a second-quadrant angle. cos(2.75) Answer Finding Angle Measures in Radians When the calculator is in radian mode, it will return the decimal value of the radian measure whose function value is entered. The measure will not be in terms of.

10 408 More Trigonometric Functions 80 For instance, we know that tan Since , an angle of 45 is an angle of 4 radians. Use a calculator to find u in radians when tan u5.your calculator should be in radian mode. ENTER: 2nd TAN ) ENTER DISPLAY: tan - () The value returned by the calculator is aroximately equal to 4. We can verify this by dividing by 4 and comaring the answers. The exact radian measure of the angle is the irrational number 4. The rational aroximation of the radian measure of the angle is EXAMPLE 3 Find, in radians, the smallest ositive value of u if tan u Exress the answer to the nearest hundredth. Solution The calculator should be in radian mode. ENTER: 2nd TAN ) DISPLAY: ENTER Tangent is negative in the second and fourth quadrants. The calculator has returned the measure of the negative rotation in the fourth quadrant. The angle in the second quadrant with a ositive measure is formed by the oosite ray and can be found by adding, the measure of a straight angle, to the given measure. tan - ( ) y x O.8762 ENTER: 2nd ANS 2nd DISPLAY: Ans+π ENTER Check Verify this answer by finding tan ENTER: TAN ENTER 2nd ANS ) DISPLAY: tan(ans) Answer 2.06

11 Trigonometric Function Values and Radian Measure 409 EXAMPLE 4 Find, to the nearest ten-thousandth, a value of u in radians if csc u Solution If csc u52.369, then sin u ENTER: 2nd SIN DISPLAY: sin - (/2.369) ) ENTER Rounded to four decimal laces, u Answer Exercises Writing About Mathematics. Alexia said that when u is a second-quadrant angle whose measure is in radians, the measure of the reference angle in radians is 2u. Do you agree with Alexia? Exlain why or why not. 2. Diego said that when u is the radian measure of an angle, the angle whose radian measure is 2n uis an angle with the same terminal side for all integral values of n. Do you agree with Diego? Exlain why or why not. Develoing Skills In 3 2, find the exact function value of each of the following if the measure of the angle is given in radians. 3. sin 4 4. tan 3 5. cos 2 6. tan cos sin tan 4 0. sec 3. csc 2. cot 4 In 3 24, find, to the nearest ten-thousandth, the radian measure u of a first-quadrant angle with the given function value. 3. sin u cos u tan u cos u sin u cos u tan u cot u sec u cot u csc u cot u If f(x) 5, find. 26. If f(x) 5 cos 2x, find. 27. If f(x) 5 sin 2x cos 3x, find. 28. If f(x) 5 tan 5x 2 sin 2x, find. Alying Skills sin A 3 x B f A 2 B f A 4 B f A 3 4 B 29. The unit circle intersects the x-axis at R(, 0) and the terminal side of ROP at P. What are the coordinates of P if mrpx ? f A 6 B

12 40 More Trigonometric Functions 30. The x-axis intersects the unit circle at R(, 0) and a circle y of radius 3 centered at the origin at A(3, 0). The terminal B side of ROP intersects the unit circle at P and the circle P of radius 3 at B. The measure of RPX is A(3, 0) a. What is the radian measure of ROP? b. What is the radian measure of AOB? c. What are the coordinates of P? d. What are the coordinates of B? e. In what quadrant do P and B lie? Justify your answer. O R(, 0) x 3. The wheels of a cart that have a radius of 2 centimeters move in a counterclockwise direction for 20 meters. a. What is the radian measure of the angle through which each wheel has turned? b. What is sine of the angle through which the wheels have turned? 32. A suorting cable runs from the ground to the to of a tree that is in danger of falling 2 down. The tree is 8 feet tall and the cable makes an angle of 9 with the ground. Determine the length of the cable to the nearest tenth of a foot. 33. An airlane climbs at an angle of 5 with the ground. When the airlane has reached an altitude of 500 feet: a. What is the distance in the air that the airlane has traveled? b. What is the horizontal distance that the airlane has traveled? Hands-On Activity :The Unit Circle and Radian Measure In Chater 9, we exlored trigonometric function values on the unit circle with regard to degree measure. In the following Hands-On Activity, we will examine trigonometric function values on the unit circle with regard to radian measure.. Recall that for any oint P on the unit circle, OP h is the terminal side of an angle in standard osition. If the measure of this angle is u, the coordinates of P are (cos u, sin u). Use your knowledge of the exact cosine and sine values for 0,30,45,60, and 90 angles to find the coordinates of the first-quadrant oints. 2 (cos, sin ) 3 (cos 4, sin 3 4 ) 5 (cos, sin ) (cos, sin ) 7 (cos, sin ) 5 (cos, sin ) 4 (cos 3, sin 4 3 ) y (cos 2, sin 2 ) ( cos 3, sin 3 ) 2 3 (cos, sin ) (cos 6, sin 6 ) O x (cos 0, sin 0) (cos, sin 6 6 ) 7 (cos, sin 4 6 ) 5 (cos, sin ) 3 (cos, sin )

13 2. The cosine and sine values of the other angles marked on the unit circle can be found by relating them to the oints in the first quadrant or along the ositive axes. For examle, the oint is the image of the oint under a reflection in the A cos 2 3, sin 2 3 B A cos 3, sin 3 B 5 (a, b), then A cos 2 3, sin 2 3 B y-axis. Thus, if A cos 3, sin 3 B Pythagorean Identities 4 5 (2a, b). Determine the exact coordinates of the oints shown on the unit circle to find the function values. Hands-On Activity 2: Evaluating the Sine and Cosine Functions The functions sin x and cos x can be reresented by the following series when x is in radians: sin x 5 x 2 x3 3! x5 5! 2 x7 7! c cos x 5 2 x2 2! x4 4! 2 x6 6! c. Write the next two terms for each series given above. 2. Use the first six terms of the series for sin x to aroximate sin 4 to four decimal laces. 3. Use the first six terms of the series for cos x to aroximate cos 3 to four decimal laces. 0-3 PYTHAGOREAN IDENTITIES The unit circle is a circle of radius with center at the origin. Therefore, the equation of the unit circle is x 2 y or x 2 y 2 5. Let P be any oint on the unit circle and OP h be the terminal side of ROP, an angle in standard osition whose measure is u.the coordinates of P are (cos u, sin u). Since P is a oint on the circle whose equation is x 2 y 2 5, and x 5 cos u and y 5 sin u: (cos u) 2 (sin u) 2 5 In order to emhasize that it is the cosine value and the sine value that are being squared, we write (cos u) 2 as cos 2 u and (sin u) 2 as sin 2 u. Therefore, the equation is written as: cos 2 usin 2 u 5 This equation is called an identity. x R(, 0) Because the identity cos 2 usin 2 u5 is based on the Pythagorean theorem, we refer to it as a Pythagorean identity. y O u P(cos u, sin u) DEFINITION An identity is an equation that is true for all values of the variable for which the terms of the variable are defined.

14 42 More Trigonometric Functions EXAMPLE Verify that cos 2 3 sin Solution We know that cos and sin 3 5!3 2. Therefore: cos 2 usin 2 u5 Q 2 R2 Q!3 2 R We can write two related Pythagorean identities by dividing both sides of the equation by the same exression. Divide by cos 2 u: sin u Recall that tan u5cos u and sec u5 cos u. For all values of u for which cos u 0: cos 2 usin 2 u5 cos 2 u cos 2 u sin2 u cos 2 u 5 cos 2 u A sin u 5 A cos u B 2 cos u B 2 tan 2 u5sec 2 u Divide by sin 2 u: Recall that cot u5tan u 5 cos sin u u and csc u5 sin u. For all values of u for which sin u 0: cos 2 usin 2 u5 cos 2 u sin 2 u sin2 u sin 2 u 5 A cos u sin u B 2 sin 2 u 5 A sin u B 2 tan 2 u 5 sec 2 u cot 2 u 5 csc 2 u EXAMPLE 2 If cos u53 and u is in the fourth quadrant, use an identity to find: a. sin u b. tan u c. sec u d. csc u e. cot u Solution Since u is in the fourth quadrant, cosine and its recirocal, secant, are ositive; sine and tangent and their recirocals, cosecant and cotangent, are negative.

15 Pythagorean Identities 43 a. cos 2 usin 2 u5 A 3 B 2 9 sin 2 u5 sin 2 u5 sin 2 u5 sin u52# 8 9 sin u !2 3 sin u b. tan u5cos u 5 2 2!2 3 3 c. sec u5 5 cos u !2 d. csc u5 sin u !2 523! !2 3 e. cot u5tan u 5 22!2 52!2 4 Answers a. 2 2!2 3 b. 22!2 c. 3 d. 2 3!2 4 e. 2!2 4 EXAMPLE 3 Write tan ucot u in terms of sin u and cos u and write the sum as a single fraction in simlest form. sin u cos u Solution tan ucot u5cos u sin u sin u cos u 5 cos u 3 sin sin u u sin u 3 cos cos u u sin cos 5 2 u 2 u cos u sin u cos u sin u sin 5 2 ucos 2 u cos u sin u 5 cos u sin u Answer SUMMARY Recirocal Identities Quotient Identities Pythagorean Identities sec u5cos u sin u tan u5cos u cos 2 usin 2 u5 csc u5sin u cos u cot u5 sin u tan 2 u5sec 2 u cot u 5 tan u cot 2 u 5 csc 2 u Exercises Writing About Mathematics. Emma said that the answer to Examle 3 can be simlified to (sec u)(csc u). Do you agree with Emma? Justify your answer. 2. Ethan said that the equations cos 2 u5 2 sin 2 u and sin 2 u5 2 cos 2 u are identities. Do you agree with Ethan? Exlain why or why not.

16 44 More Trigonometric Functions Develoing Skills In 3 4, for each given function value, find the remaining five trigonometric function values. 3. sin u5 5 and u is in the second quadrant cos u54 and u is in the first quadrant. 5. cos u and u is in the third quadrant. 6. sin u and u is in the fourth quadrant sin u5 3 and u is in the second quadrant. 8. tan u 522 and u is in the second quadrant. 9. tan u 54 and u is in the third quadrant. 0. sec u528 and u is in the second quadrant. 5. cot u5 3 and u is in the third quadrant csc u54 and u is in the second quadrant sin u5 5 and u is in the second quadrant. 4. cot u 526 and u is in the fourth quadrant. In 5 22, write each given exression in terms of sine and cosine and exress the result in simlest form. 5. (csc u)(sin u) 6. (sin u)(cot u) 7. (tan u)(cos u) 8. (sec u)(cot u) 9. sec utan u sec u 20. csc u 2. csc 2 cot u u2 tan u 22. sec u ( cot u) 2 csc u ( tan u) 0-4 DOMAIN AND RANGE OF TRIGONOMETRIC FUNCTIONS In Chater 9, we defined the trigonometric functions for degree measures on the unit circle. Using arc length, we can define the trigonometric functions on the set of real numbers. Recall that the function values of sine and cosine are given by the coordinates of P, that is, P(cos u, sin u). The length of the arc interceted by the angle is given by P(cos u, sin u) u5 s r or s 5ur However, since the radius of the unit circle is, the length of the arc s is equal to u. In other words, the length of the arc s is equivalent to the measure of the angle u in radians, a real number. Thus, any trigonometric function of u is a function on the set of real numbers. y O u s 5 ur R(, 0) x

17 Domain and Range of Trigonometric Functions 45 The Sine and Cosine Functions We have defined the sine function and the cosine function in terms of the measure of angle formed by a rotation. The measure is ositive if the rotation is in the counterclockwise direction and negative if the rotation is in the clockwise rotation. The rotation can continue indefinitely in both directions. The domain of the sine function and of the cosine function is the set of real numbers. To find the range, let P be any oint on the unit circle and OP h be the terminal side of ROP, an angle in standard osition with measure u. The coordinates of P are (cos u, sin u). Since the unit circle has a radius of, as P moves on the circle, its coordinates are elements of the interval [2, ]. Therefore, the range of cos u and sin u is the set of real numbers [2, ]. To see this algebraically, consider the identity cos 2 usin 2 u5. Subtract cos 2 u from both sides of this identity and solve for sin u. Similarly, subtract sin 2 u from both sides of this identity and solve for cos u. x P y O u R(, 0) Q cos 2 usin 2 u5 cos 2 usin 2 u5 sin 2 u5 2 cos 2 u cos 2 u5 2 sin 2 u sin u56" 2 cos 2 u cos u56" 2 sin 2 u The value of sin u will be a real number if and only if 2 cos 2 u$0, that is, if and only if cos u # or 2 # cos u #. The value of cos u will be a real number if and only if 2 sin 2 u$0, that is, if and only if sin u # or 2 # sin u #. The range of the sine function and of the cosine function is the set of real numbers [2, ], that is, the set of real numbers from 2 to, including 2 and. The Secant Function The secant function is defined for all real numbers such that cos u 0.We know 3 that cos u50 when u5 2, when u5 2, and when u differs from either of these values by a comlete rotation, 2.

18 46 More Trigonometric Functions Therefore, cos u 5 0 when u equals: ( ( We can continue with this attern, which can be summarized as 2 n for all integral values of n. The domain of the secant function is the set of all real numbers excet those for which cos u50. The domain of the secant function is the set of real numbers excet 2 n for all integral values of n. The secant function is defined as sec u5 cos u. Therefore: When 0, cos u #, 0 cos u, cos cos u u # cos u When 2 # cos u,0, 2 cos u $ cos cos u u. cos 0 u or 0, # sec u or 2sec u $. 0 or sec u #2, 0 The union of these two inequalities defines the range of the secant function. The range of the secant function is the set of real numbers (2`, 2] : [, `), that is, the set of real numbers greater than or equal to or less than or equal to 2. The Cosecant Function The cosecant function is defined for all real numbers such that sin u 0. We know that sin u 50 when u 50, when u 5, and when u differs from either of these values by a comlete rotation, 2. Therefore, sin u 50 when u equals: ( ( We can continue with this attern, which can be summarized as n for all integral values of n. The domain of the cosecant function is the set of all real numbers excet those for which sin u 50.

19 Domain and Range of Trigonometric Functions 47 The domain of the cosecant function is the set of real numbers excet n for all integral values of n. The range of the cosecant function can be found using the same rocedure as was used to find the range of the secant function. The range of the cosecant function is the set of real numbers (2`, 2] : [, `), that is, the set of real numbers greater than or equal to or less than or equal to 2. The Tangent Function sin u We can use the identity tan u5cos u to determine the domain of the tangent function. Since sin u and cos u are defined for all real numbers, tan u is defined for all real numbers for which cos u 0, that is, for all real numbers excet 2 n. The domain of the tangent function is the set of real numbers excet 2 n for all integral values of n. To find the range of the tangent function, recall that the tangent function is defined on the line tangent to the unit circle at R(, 0). In articular, let T be the oint where the terminal side of an angle in standard osition intersects the line tangent to the unit circle at R(, 0). If m ROT 5u, then the coordinates of T are (, tan u).as the oint T moves on the tangent line, its y-coordinate can take on any real-number value. Therefore, the range of tan u is the set of real numbers. To see this result algebraically, solve the identity tan 2 u5sec 2 u for tan 2 u. tan 2 u5sec 2 u tan 2 u5sec 2 u2 The range of the secant function is sec u $ or sec u #2. Therefore, sec 2 u$. If we subtract from each side of this inequality: sec 2 u2 $ 0 tan 2 u$0 tan u $0 or tan u #0 The range of the tangent function is the set of all real numbers. y O u P T(, tan u) x R(, 0)

20 48 More Trigonometric Functions The Cotangent Function We can find the domain and range of the cotangent function by a rocedure similar to that used for the tangent function. cos u We can use the identity cot u5sin u to determine the domain of the cotangent function. Since sin u and cos u are defined for all real numbers, cot u is defined for all real numbers for which sin u 0, that is, for all real numbers excet n. The domain of the cotangent function is the set of real numbers excet n for all integral values of n. To find the range of the cotangent function, solve the identity cot 2 u5csc 2 u for cot 2 u. cot 2 u5csc 2 u cot 2 u5csc 2 u2 The range of the cosecant function is csc u $ or csc u #2. Therefore, csc 2 u$. If we subtract from each side of this inequality: csc 2 u2 $ 0 cot 2 u$0 cot u $0 or cot u #0 The range of the cotangent function is the set of all real numbers. EXAMPLE Exlain why that is no value of u such that cos u 52. Solution The range of the cosine function is 2 # cos u #. Therefore, there is no value of u for which cos u is greater than. SUMMARY Function Domain (n is an integer) Range Sine All real numbers [2, ] Cosine All real numbers [2, ] Tangent All real numbers excet 2 n All real numbers Cotangent All real numbers excet n All real numbers Secant All real numbers excet 2 n (2`, 2] < [, `) Cosecant All real numbers excet n (2`, 2] < [, `)

21 Inverse Trigonometric Functions 49 Exercises Writing About Mathematics. Nathan said that cot u is undefined for the values of u for which csc u is undefined. Do you agree with Nathan? Exlain why or why not. 2. Jonathan said that cot u is undefined for 2 because tan u is undefined for 2. Do you agree with Nathan? Exlain why or why not. Develoing Skills In 3 8, for each function value, write the value or tell why it is undefined. Do not use a calculator. 3. sin 2 4. cos 2 5. tan 2 6. sec 2 7. csc 2 8. cot 2 9. tan 0. cot. sec csc tan 0 4. cot cot (28) tan A 2 2 B csc A B 9. List five values of u for which sec u is undefined. 20. List five values of u for which cot u is undefined. sec A B 0-5 INVERSE TRIGONOMETRIC FUNCTIONS Inverse Sine Function We know that an angle of 30 is an angle of 6 radians and that an angle of 5(30 ) 5 or 50 is an angle of 5 A 6 B or 6. Since , an angle of 30 is the reference angle for an angle of 50. Therefore: sin 30 5 sin and sin 6 5 sin A 6, 2 B Two ordered airs of the sine function are and. Therefore, the sine function is not a one-to-one function. Functions that are not one-to-one do not have an inverse function. When we interchange the elements of the ordered airs of the sine function, the set of ordered airs is a relation that is not a function. {(a, b) : b 5 sin a} is a function. {(b, a) : a 5 arcsin b} is not a function. A 5 6, 2 B

22 420 More Trigonometric Functions For every value of u such that 0 # u# 2, there is exactly one nonnegative value of sin u. For every value of u such that 2 2 #u,0, that is, for every fourth-quadrant angle, there is exactly one negative value of sin u. Therefore, if we restrict the domain of the sine function to 2 2 #u# 2, the function is one-to-one and has an inverse function. We designate the inverse sine function by arcsin or sin 2. y 2 O 2 2 x Sine Function with a Restricted Domain y 5 sin x Domain 5 Ux : 2 2 # x # 2 V Range 5 {y : 2 # y # } Inverse Sine Function y 5 arcsin x or y 5 sin 2 x Domain 5 {x : 2 # x # } Range 5 Uy : 2 2 # y # 2 V Note that when we use the SIN key on the grahing calculator with a number in the interval [2, ], the resonse will be an number in the interval S2 2, 2 T. Inverse Cosine Function The cosine function, like the sine function, is not a one-to-one function and does not have an inverse function.we can also restrict the domain of the cosine function to form a one-to-one function that has an inverse function. For every value of u such that 0 # u# 2, y there is exactly one nonnegative value of cos u. For every value of u such that 2,u#, that is, for every second-quadrant angle, there is exactly one negative value of cos u. Therefore, if we restrict the domain of the cosine function to 0 #u#, the function is one-to-one and has an inverse function. We designate the inverse cosine function by arccos or cos 2. O 0 x Cosine Function with a Restricted Domain y 5 cos x Domain 5 {x :0 # x # } Range 5 {y : 2 # y # } Inverse Cosine Function y 5 arccos x or y 5 cos 2 x Domain 5 {x : 2 # x # } Range 5 {y :0# y #}

23 Inverse Trigonometric Functions 42 Note that when we use the COS key on the grahing calculator with a number in the interval [2, ], the resonse will be an number in the interval [0, ]. Inverse Tangent Function The tangent function, like the sine and cosine functions, is not a one-to-one function and does not have an inverse function. We can also restrict the domain of the tangent function to form a one-to-one function that has an inverse function. For every of u such that 0 # u, 2, there is exactly one nonnegative value of tan u. For every value of u such that 2 2,u,0, that is, for every fourth-quadrant angle, there is exactly y 2 one negative value of tan u. Therefore, if we O restrict the domain of the tangent function to x 2 2,u, 2, the function is one-to-one and has an inverse function. We designate the inverse 2 tangent function by arctan or tan 2. 2 Tangent Function with a Restricted Domain y 5 tan x Domain 5 Ux : 2 2, x, 2 V Range 5 {y : y is a real number} Inverse Tangent Function y 5 arctan x or y 5 tan 2 x Domain 5 {x : x is a real number} Range 5 Uy : 2 2, y, 2 V Note that when we use the TAN key on the grahing calculator with any real number, the resonse will be an number in the interval A 2 2, 2 B. These same restrictions can be used to define inverse functions for the secant, cosecant, and cotangent functions, which will be left to the student. (See Exercises ) EXAMPLE Exress the range of the arcsine function in degrees. Solution Exressed in radians, the range of the arcsine function is S2, that is, 2 2, 2 T 2 # arcsin u # 2. Since an angle of 2 radians is an angle of 90 and an angle of 2 2 radians is an angle of 290, the range of the arcsine function is: [290,90 ] or 290 #u#90 Answer

24 422 More Trigonometric Functions EXAMPLE 2 Find sin (cos 2 (20.5)). Solution Let u 5cos 2 (20.5). Then u is a second-quadrant angle whose reference angle, r, is r 52uin radians or r uin degrees. In radians: In degrees: cos r cos r r 5 3 r u u u 20 5u!3 sin u5sin sin u5sin 20 5!3 2 Alternative Solution To find sin (cos 2 (20.5)), find sin u when u5cos 2 (20.5). If u5cos 2 (20.5), then cos u For the function y 5 cos 2 x, when x is negative, 2, y #. Therefore, u is the measure of a second-quadrant angle and sin u is ositive. Use the Pythagorean identity: sin 2 ucos 2 u5 sin 2 u(20.5) 2 5 sin 2 u sin 2 u50.75 sin u5!0.75 5!0.25! !3 Check ENTER: SIN 2nd COS (-) DISPLAY: sin(cos - ( -.5)).5 ) ) ENTER nd 3 ) 2 (3)/ ENTER Answer!3 2 EXAMPLE 3 Find the exact value of sec A arcsin 23 5 B if the angle is a third-quadrant angle. Solution Let u5 A arcsin 23 5 B. This can be written as sin u5 A 23 5 B. Then: sec A arcsin 23 5 B 5 sec u5 cos u

25 Inverse Trigonometric Functions 423 We know the value of sin u. We can use a Pythagorean identity to find cos u: cos u56" 2 sin 2 u cos u 5 5 6" 2 sin 2 u 6# 2 A B 2 6# # 6 25 An angle in the third quadrant has a negative secant function value Answer sec A arcsin 23 5 B 524 Calculator Solution ENTER: COS 2nd 23 5 ) ) ENTER ENTER SIN MATH DISPLAY: /cos(sin - ( - 3/5) ) Frac 5/4 5 The calculator returns the value. The calculator used the function value of A arcsin B, a fourth-quadrant angle for which the secant function value is ositive. For the given third-quadrant angle, the secant function value is negative. 5 Answer sec A arcsin 23 5 B 524 Exercises Writing About Mathematics. Nicholas said that the restricted domain of the cosine function is the same as the restricted domain of the tangent function. Do you agree with Nicholas? Exlain why or why not. 2. Sohia said that the calculator solution to Examle 2 could have been found with the calculator set to either degree or radian mode. Do you agree with Sohia? Exlain why or why not. Develoing Skills In 3 4, find each value of u: a. in degrees b. in radians 3. u5arcsin 2 4. u5arctan 5. u5arccos 6. u5arctan (2) 7. u5arccos A 2 2 B 8. u5arcsin A 2 2 B 9. u5arctan 0. u5arcsin Q2!3 2 R. u5arccos (2) 2. u5arcsin 3. u5arctan 0 4. u5arccos 0

26 424 More Trigonometric Functions In 5 23, use a calculator to find each value of u to the nearest degree. 5. u5arcsin u5arccos (20.6) 7. u5arctan u5arctan (24.4) 9. u5arccos u5arccos (20.9) 2. u5arcsin u5arcsin (20.72) 23. u5arctan (27.3) In 24 32, find the exact value of each exression. 24. sin (arctan ) 25. cos (arctan 0) 26. tan (arccos ) 27. cos (arccos (2)) 28. tan 29. cos 30. tan Qarccos Q2!2 2 RR 3. sin Qarccos Q2!2 2 RR 32. cos A arcsin A 2 2 BB Qarcsin Q2!3 2 RR Qarcsin Q2!2 2 RR In 33 38, find the exact radian measure u of an angle with the smallest absolute value that satisfies the equation.!2!3 33. sin u cos u tan u5 36. sec u csc u52!2 38. cot u5!3 39. In a c with the triangles labeled as shown, use the inverse trigonometric functions to exress u in terms of x. a. b. c. x 2x 3 3 x u x u u x 2 Alying Skills 40. a. Restrict the domain of the secant function to form a one-to-one function that has an inverse function. Justify your answer. b. Is the restricted domain found in a the same as the restricted domain of the cosine function? c. Find the range of the restricted secant function. d. Find the domain of the inverse secant function, that is, the arcsecant function. e. Find the range of the arcsecant function.

27 4. a. Restrict the domain of the cosecant function to form a one-to-one function that has an inverse function. Justify your domain. b. Is the restricted domain found in a the same as the restricted domain of the sine function? c. Find the range of the restricted cosecant function. d. Find the domain of the inverse cosecant function, that is, the arccosecant function. e. Find the range of the arccosecant function. 42. a. Restrict the domain of the cotangent function to form a one-to-one function that has an inverse function. Justify your domain. b. Is the restricted domain found in a the same as the restricted domain of the tangent function? c. Find the range of the restricted cotangent function. d. Find the domain of the inverse cotangent function, that is, the arccotangent function. e. Find the range of the arccotangent function. 43. Jennifer lives near the airort. An airlane aroaching the airort flies at a constant altitude of mile toward a oint, P, above Jennifer s house. Let u be the measure of the angle of elevation of the lane and d be the horizontal distance from P to the airlane. a. Exress u in terms of d. b. Find u when d 5 mile and when d mile. Cofunctions 425 d u P 0-6 COFUNCTIONS Two acute angles are comlementary if the sum of their measures is 90. In the diagram, ROP is an angle in standard osition whose measure is u and (, q) are the coordinates of P, the intersection of OP h with the unit circle. Under a reflection in the line y = x, the image of (x, y) is (y, x). Therefore, under a reflection in the line y = x, the image of P(, q) is P (q, ), the image of R(, 0) is R (0, ), and the image of O(0, 0) is O(0, 0). Under a line reflection, angle measure is reserved. Therefore, m ROP 5 m R OP 5u m ROP m R OP u y R9(0, ) P9(q, ) y 5 x u O u P(, q) R(, 0)

28 426 More Trigonometric Functions A The two angles, ROP and ROP are in standard osition. Therefore, cos u 5 sin (90 2u) 5 sin u 5q cos (90 2u) 5 q cos u 5 sin (90 2 u) sin u 5cos (90 2 u) The sine of an acute angle is equal to the cosine of its comlement. The sine and cosine functions are called cofunctions. There are two other airs of cofunctions in trigonometry. sin u cos u tan u5 cot u5sin u 5 cos u 5 q q cos (908 2u) sin (908 2u) cot (90 2u) 5 tan (90 2u) 5 cos (908 2u) 5 sin (908 2u) 5 q q tan u 5 cot (90 2 u) cot u 5 tan (90 2 u) The tangent and cotangent functions are cofunctions. sec u5cos u 5 csc u5sin u 5 q csc (90 2u) 5 sin (908 2u) 5 sec (90 2u) 5 cos (908 2u) 5 q sec u 5 csc (90 2 u) csc u 5 sec (90 2 u) The secant and cosecant functions are cofunctions. Cofunctions reresented in radians are left to the student. (See Exercise 25.) In any right triangle ABC, if C is the right angle, then A and B are comlementary angles. Therefore, m A m B and m B m A. C B Sine and cosine are cofunctions: sin A 5 cos B and sin B 5 cos A. Tangent and cotangent are cofunctions: tan A 5 cot B and tan B 5 cot A. Secant and cosecant are cofunctions: sec A 5 csc B and sec B 5 csc A. EXAMPLE If tan , find a value of u such that cot u Solution Answer If u is the degree measure of a first-quadrant angle, then: tan (90 2u) 5 cot u tan cot u Therefore, tan (90 2u) 5 tan and (90 2u) u u u526.56

29 Cofunctions 427 EXAMPLE 2 Exress sin 00 as the function value of an acute angle that is less than 45. Solution An angle whose measure is 00 is a second-quadrant angle whose reference angle is or 80. Therefore, sin 00 5 sin 80 5 cos ( ) 5 cos 0. Answer sin 00 5 cos 0 Exercises Writing About Mathematics. Mia said that if you know the sine value of each acute angle, then you can find any trigonometric function value of an angle of any measure. Do you agree with Mia? Exlain why or why not.!3 2. If sin A 5 2, is cos A 5 2 always true? Exlain why or why not. Develoing Skills In 3 22: a. Rewrite each function value in terms of its cofunction. b. Find, to four decimal laces, the value of the function value found in a. 3. sin cos tan sin csc sec cot cos 70. sin 0 2. tan cos sec sin cos tan csc cos sin cot sec If sin u 5cos (20 u), what is the value of u? 24. For what value of x does tan (x 0) 5 cot (40 x)? 25. Comlete the following table of cofunctions for radian values. Cofunctions (degrees) cos u5sin (90 2u) sin u5cos (90 2u) tan u5cot (90 2u) cot u5tan (90 2u) sec u5csc (90 2u) csc u5sec (90 2u) Cofunctions (radians)

30 428 More Trigonometric Functions In 26 33: a. Rewrite each function value in terms of its cofunction. b. Find the exact value of the function value found in a. 26. sin cos tan sec csc 6 3. cot 32. sin A 2 4 B cos tan A B CHAPTER SUMMARY A radian is the unit of measure of a central angle that intercets an arc equal in length to the radius of the circle. In general, the radian measure u of a central angle is the length of the interceted arc, s, divided by the radius of the circle, r. u5 s r The length of the semicircle is r. Therefore, the radian measure of a r straight angle is r 5. This leads to the relationshi: radians 5 80 degrees The measure of any angle can be exressed in degrees or in radians. We can do this by forming the following roortion: measure in degrees measure in radians 5 80 When the calculator is in radian mode, it will return the decimal value of the radian measure whose function value is entered. The measure will not be in terms of. An identity is an equation that is true for all values of the variable for which the terms of the variable are defined. Recirocal Identities Quotient Identities Pythagorean Identities sec u5cos u sin u tan u5cos u cos 2 usin 2 u5 csc u5sin u cos u cot u5 sin u tan 2 u5sec 2 u cot u5tan u cot 2 u 5 csc 2 u

31 Chater Summary 429 The domains and ranges of the six basic trigonometric functions are as follows. Function Domain (n is an integer) Range Sine All real numbers [2, ] Cosine All real numbers [2, ] Tangent All real numbers excet 2 n All real numbers Cotangent All real numbers excet n All real numbers Secant All real numbers excet 2 n (2`, 2] < [, `) Cosecant All real numbers excet n (2`, 2] < [, `) The inverse trigonometric functions are defined for restricted domains. Sine Function with a Restricted Domain y 5 sin x Domain 5 Ux : 2 2 # x # 2 V Range 5 {y : 2 # y # } Inverse Sine Function y 5 arcsin x or y 5 sin 2 x Domain 5 {x : 2 # x # } Range 5 Uy : 2 2 # y # 2 V Cosine Function with a Restricted Domain y 5 cos x Domain 5 {x :0 # x # } Range 5 {y : 2 # y # } Inverse Cosine Function y 5 arccos x or y 5 cos 2 x Domain 5 {x : 2 # x # } Range 5 {y :0# y #} Tangent Function with a Restricted Domain y 5 tan x Domain 5 Ux : 2 2, x, 2 V Range 5 {y : y is a real number} Inverse Tangent Function y 5 arctan x or y 5 tan 2 x Domain 5 {x : x is a real number} Range 5 Uy : 2 2, y, 2 V A C B In any right triangle ABC, if C is the right angle, then A and B are comlementary angles. Therefore, m A m B and m B m A. Sine and cosine are cofunctions: sin A 5 cos B and sin B 5 cos A. Tangent and cotangent are cofunctions: tan A 5 cot B and tan B 5 cot A. Secant and cosecant are cofunctions: sec A 5 csc B and sec B 5 csc A.

32 430 More Trigonometric Functions VOCABULARY 0- Radian 0-3 Identity Pythagorean identity 0-5 Restricted domain Inverse sine function Inverse cosine function Inverse tangent function 0-6 Cofunction REVIEW EXERCISES In 4, exress each degree measure in radian measure in terms of In 5 8, exress each radian measure in degree measure What is the radian measure of a right angle? In 0 7, in a circle of radius r, the measure of a central angle is u and the length of the arc interceted by the angle is s. 0. If r 5 5 cm and s 5 5 cm, find u.. If r 5 5 cm and u 52, find s. 2. If r 5 2 in. and s 5 3 in., find u. 3. If s 5 8 cm and u 52, find r. 4. If s 5 9 cm and r 5 3 mm, find u. 5. If s 5cm and u5 4, find r. 6. If u53 and s cm, find r. 7. If u5 5 and r 5 0 ft, find s. 8. For what values of u (0 #u#360 ) is each of the following undefined? a. tan u b. cot u c. sec u d. csc u 9. What is the domain and range of each of the following functions? a. arcsin u b. arccos u c. arctan u In 20 27, find the exact value of each trigonometric function. 20. sin 6 2. cos tan sin cos tan sin cos A 27 6 B

33 Cumulative Review 43 In 28 33, find the exact value of each exression. 28. tan 3 4 cot tan sec cot 2 A 3 B sec 2 A 4 B 2 csc 2 5 cot 2 tan 2 A 7 6 B 2 sec 2 A 7 6 B 3 cot A 4 B? 2 csc 2 A 4 B In 34 39, find the exact value of each exression. 34. sin Qarcsin2!3 2 R 35. tan Aarccot 2!3B 36. cot (arccos 2) 37. sec Qarcsin!2 2 R 38. csc (arccot 2) 39. cos (arcsec 2) P y u O Q(0, ) x R(, 0) Exloration In this exloration, we will continue the work begun in Chater 9. Let OP h intersect the unit circle at P. The line that is tangent to the circle at g R(, 0) intersects OP at T. The line that is tangent to the circle at Q(0, ) intersects OP g at S. The measure of ROP is u.. ROP is a second-quadrant angle as shown to the left. Locate oints S and T. Show that 2OT 5 sec u, OS 5 csc u, and 2QS 5 cot u. 2. Draw ROP as a third-quadrant angle. Let R(, 0) and Q(0, ) be fixed oints. Locate oints S and T. Show that 2OT 5 sec u, 2OS 5 csc u, and QS 5 cot u. 3. Draw ROP as a fourth-quadrant angle. Let R and Q be defined as before. Locate oints S and T. Show that OT sec u, OS csc u, and QS cot u. CUMULATIVE REVIEW CHAPTERS 0 Part I Answer all questions in this art. Each correct answer will receive 2 credits. No artial credit will be allowed.. If sec u5!5, then cos u equals 2!5!5 () 2!5 (2) (3) (4) 2! Which of the following is an irrational number? () 0.34 (2)!2 (3) 3 2i (4) 2

34 432 More Trigonometric Functions 3. Which of the following is a function from {, 2, 3, 4, 5} to {, 2, 3, 4, 5} that is one-to-one and onto? () {(, 2), (2, 3), (3, 4), (4, 5), (5, 5)} (3) {(, 3), (2, 3), (3, 3), (4, 3), (5, 3)} (2) {(, 5), (2, 4), (3, 3), (4, 2), (5, )} (4) {(, ), (2, 4), (3, ), (4, 2), (5, )} 4. When written with a rational denominator, is equal to 3!5 3!5 3 2!5 3 2!5 3!5 () 4 (2) 22 (3) 4 (4) 8 2 a 5. The fraction is equal to 2 a 2 a () a 2 a (2) a 2 a (3) a a (4) a 6. The inverse function of the function y 5 2x is () y 5 2 x x 2 (3) y 5 2 x (2) y 5 2 (4) y 522x 2 7. The exact value of cos 2 3 is!3 () 2 (2) 2!3 2 (3) 2 (4) The series ??? written in sigma notation is ` () a 22k (3) a (k 4k) k5 ` k5 ` (2) a (22 2k) (4) a (22 4(k 2 )) k5 9. What are the zeros of the function y 5 x 3 3x 2 2x? () 0 only (3) 0,, and 2 (2) 22, 2, and 0 (4) The function has no zeros. 0. The roduct A!28BA!22B is equal to () 4i (2) 4 (3) 24 (4) 3i!2 ` k5 Part II Answer all questions in this art. Each correct answer will receive 2 credits. Clearly indicate the necessary stes, including aroriate formula substitutions, diagrams, grahs, charts, etc. For all questions in this art, a correct numerical answer with no work shown will receive only credit.. Exress the roots of x 2 2 6x in a + bi form. 2. If 280 is the measure of an angle in degrees, what is the measure of the angle in radians?

35 Cumulative Review 433 Part III Answer all questions in this art. Each correct answer will receive 4 credits. Clearly indicate the necessary stes, including aroriate formula substitutions, diagrams, grahs, charts, etc. For all questions in this art, a correct numerical answer with no work shown will receive only credit. 3. What is the equation of a circle if the endoints of the diameter are (22, 4) and (6, 0)? 4. Given log log 3 27 log 3 243: a. Write the exression as a single logarithm. b. Evaluate the logarithm. Part IV Answer all questions in this art. Each correct answer will receive 6 credits. Clearly indicate the necessary stes, including aroriate formula substitutions, diagrams, grahs, charts, etc. For all questions in this art, a correct numerical answer with no work shown will receive only credit. 5. If sec u5!3 and u is in the fourth quadrant, find cos u, sin u, and tan u in simlest radical form. 6. Find grahically the common solutions of the equations: y 5 x 2 2 4x y 5 x 2 3