TRIGONOMETRIC EQUATIONS

Transcription

1 CHAPTER 3 CHAPTER TABLE OF CONTENTS 3- First-Degree Trigonometric Equations 3- Using Factoring to Solve Trigonometric Equations 3-3 Using the Quadratic Formula to Solve Trigonometric Equations 3-4 Using Substitution to Solve Trigonometric Equations Involving More Than One Function 3-5 Using Substitution to Solve Trigonometric Equations Involving Different Angle Measures Chapter Summary Vocabulary Review Exercises Cumulative Review TRIGONOMETRIC EQUATIONS The triangle is a rigid figure, that is, its shape cannot be changed without changing the lengths of its sides. This fact makes the triangle a basic shape in construction. The theorems of geometry give us relationships among the measures of the sides and angles of triangle. Precisely calibrated instruments enable surveyors to obtain needed measurements. The identities and formulas of trigonometry enable architects and builders to formulate plans needed to construct the roads, bridges, and buildings that are an essential part of modern life. 58

2 First-Degree Trigonometric Equations FIRST-DEGREE TRIGONOMETRIC EQUATIONS A trigonometric equation is an equation whose variable is expressed in terms of a trigonometric function value. To solve a trigonometric equation, we use the same procedures that we used to solve algebraic equations. For example, in the equation 4 sin u5 5 7, sin u is multiplied by 4 and then 5 is added. Thus, to solve for sin u, first add the opposite of 5 and then divide by 4. 4 sin u sin u 5 4 sin u sin u 5 We know that sin 30 5, so one value of u is 30 ; u We also know that since sin u is positive in the second quadrant, there is a secondquadrant angle, u, whose sine is. Recall the relationship between an angle in any quadrant to the acute angle called the reference angle. The following table compares the degree measures of u from 90 to 360, the radian measures of p u from to p, and the measure of its reference angle. Fourth First Second Third Fourth Quadrant Quadrant Quadrant Quadrant Quadrant Angle 90,u,0 0,u,90 90,u,80 80,u,70 70,u,360 p 3p 3p p,u,0 0,u, p,u,p p,u,,u,p Reference u u 80 u u u Angle u u p u u p p u The reference angle for the second-quadrant angle whose sine is has a degree measure of 30 and u or 50. Therefore, sin u 5. For 0 u,360, the solution set of 4 sin u5 5 7 is {30, 50 }. In radian measure, the solution set is U p 6, 5p 6 V. In the example given above, it was possible to give the exact value of u that makes the equation true. Often it is necessary to use a calculator to find an approximate value. Consider the solution of the following equation. 5 cos u cos u54 4 cos u55 u5arccos A 4 5 B

3 50 Trigonometric Equations When we use a calculator to find u, the calculator will return the value of the function y 5 arccos x whose domain is 0 x 80 in degree measure or 0 x p in radian measure. In degree measure: ENTER: nd COS (-) 4 DISPLAY: cos - ( - 4/5) 5 ) ENTER To the nearest degree, one value of u is y 43. In addition to this second-quadrant angle, there is a third-quadrant angle such that cos u To find this third-quadrant 43 angle, find the reference angle for u. 37 Let R be the measure of the reference x angle of the second-quadrant angle. That is, R is the acute angle such that cos u5cos R. R 5 80 u The measure of the third-quadrant angle is: u5r80 u u57 For 0 u 360, the solution set of 5 cos u7 5 3 is {43, 7 }. If the value of u can be any angle measure, then for all integral values of n, u n or u57 360n. Procedure To solve a linear trigonometric equation:. Solve the equation for the function value of the variable.. Use a calculator or your knowledge of the exact function values to write one value of the variable to an acceptable degree of accuracy. 3. If the measure of the angle found in step is not that of a quadrantal angle, find the measure of its reference angle. 4. Use the measure of the reference angle to find the degree measures of each solution in the interval 0 u,360 or the radian measures of each solution in the interval 0 u,p. 5. Add 360n (n an integer) to the solutions in degrees found in steps and 4 to write all possible solutions in degrees. Add pn (n an integer) to the solutions in radians found in steps and 4 to write all possible solutions in radians.

4 First-Degree Trigonometric Equations 5 The following table will help you find the locations of the angles that satisfy trigonometric equations. The values in the table follow from the definitions of the trigonometric functions on the unit circle. Sign of a and b (0 * a *, b 0) sin u5a Quadrants I and II Quadrants III and IV cos u5a Quadrants I and IV Quadrants II and III tan u5b Quadrants I and III Quadrants II and IV EXAMPLE Find the solution set of the equation 7 tan u5!3 tan u in the interval 0 u,360. Solution How to Proceed () Solve the equation for tan u: 7 tan u5!3 tan u 6 tan u5!3 tan u 5!3 3 () Since tan u is positive, u can be a u 5 30 first-quadrant angle: (3) Since u is a first-quadrant angle, R 5 30 R 5u: (4) Tangent is also positive in the third u 5 80 R quadrant. Therefore, there is a u third-quadrant angle such that!3 tan u5 3. In the third quadrant, u 5 80 R: Answer The solution set is {30, 0 }.

5 5 Trigonometric Equations EXAMPLE Find, to the nearest hundredth, all possible solutions of the following equation in radians: 3(sin A ) 5 3 sin A Solution How to Proceed () Solve the equation for sin A: 3(sin A ) 5 3 sin A 3 sin A sin A 4 sin A 53 sin A () Use a calculator to find one value of A (be sure that the calculator is in RADIAN mode): ENTER: nd SIN (-) 3 4 DISPLAY: ENTER sin - ( - 3/4) ) (3) Find the reference angle: R 5A 5(0.848) (4) Sine is negative in quadrants In quadrant III: A 5p III and IV. Use the In quadrant IV: A 5 p reference angle to find a value of A in each of these quadrants: (5) Write the solution set: {3.99 pn, 5.44 pn} Answer A 3 4 B One value of A is Note: When sin was entered, the calculator returned the value in the interval p #u#p, the range of the inverse of the sine function. This is the measure of a fourth-quadrant angle that is a solution of the equation. However, solutions are usually given as angle measures in radians between 0 and p plus multiples of p. Note that the value returned by the calculator is 5.43 p () EXAMPLE 3 Find all possible solutions to the following equation in degrees: ( sec u3) 5 sec u5

6 First-Degree Trigonometric Equations 53 Solution () Solve the equation for sec u: () Rewrite the equation in terms of cos u: (3) Use a calculator to find one value of u: ( sec u3) 5 sec u5 sec u3 5 sec u5 cos u 5 5 sec u ENTER: nd COS DISPLAY: ENTER cos - ( - /) 0 ) Cosine is negative in quadrant II so u 5 0. (4) Find the reference angle: R (5) Cosine is also negative in In quadrant III: u quadrant III. Use the reference angle to find a value of u in quadrant III: (6) Write the solution set: {0 360n, n} Answer Trigonometric Equations and the Graphing Calculator Just as we used the graphing calculator to approximate the irrational solutions of quadratic-linear systems in Chapter 5, we can use the graphing calculator to approximate the irrational solutions of trigonometric equations. For instance, 3(sin A ) 5 3 sin A from Example can be solved using the intersect feature of the calculator. STEP. Treat each side of the equation as a function. Enter Y 5 3(sin X ) and Y 5 3 sin X into the Y menu. STEP. Using the following viewing window: p Xmin 5 0, Xmax 5 p, Xscl 5 6, Ymin 5 0, Ymax 5 0 and with the calculator set to radian mode, GRAPH the functions.

7 54 Trigonometric Equations STEP 3. The solutions are the x-coordinates of the intersection points of the graphs. We can find the intersection points by using the intersect function. Press nd CALC 5 ENTER ENTER to select both curves. When the calculator asks you for a guess, move the cursor near one of the intersection points using the arrow keys and then press ENTER. Repeat this process to find the other intersection point. * Intersection X= Y=3.75 * Intersection X= Y=3.75 As before, the solutions in the interval 0 u,p are approximately 3.99 and The solution set is {3.99 pn, 5.44 pn}. Exercises Writing About Mathematics. Explain why the solution set of the equation x is {} but the solution set of the equation sin x is { }, the empty set.. Explain why x has only one solution in the set of real numbers but the equation tan x has infinitely many solutions in the set of real numbers. Developing Skills In 3 8, find the exact solution set of each equation if 0 u, cos u tan u! sin u 5 sin u 6. 5(cos u) (tan u) 5 tan u7 8. sec u! 5! In 9 4, find the exact values for u in the interval 0 u,p sin u!35sin u 0. 5 cos u3 5 3 cos u5. tan u 5 tan u!. sin u! csc u5 5 csc u9 4. 4(cot u) 5 (cot u)

8 First-Degree Trigonometric Equations 55 In 5 0, find, to the nearest degree, the measure of an acute angle for which the given equation is true. 5. sin u3 5 5 sin u 6. 3 tan u 5 tan u cos u 5 8 cos u 8. 4(sin u) 5 6 sin u 9. csc u 5 3 csc u 0. cot u8 5 3 cot u In 4, find, to the nearest tenth, the degree measures of all u in the interval 0 u,360 that make the equation true.. 8 cos u53 4 cos u. 5 sin u 5 sin u 3. tan u4 5 3 tan u4 4. sec u55 sec u In 5 8, find, to the nearest hundredth, the radian measures of all u in the interval 0 u,p that make the equation true sin u 5 3 sin u 6. 9 cos u58 4 cos u 7. 5 tan u7 5 5 tan u3 8. cot u6 5 cot u Applying Skills 9. The voltage E (in volts) in an electrical circuit is given by the function E 5 0 cos (pt) where t is time in seconds. a. Graph the voltage E in the interval 0 t. b. What is the voltage of the electrical circuit when t 5? c. How many times does the voltage equal volts in the first two seconds? d. Find, to the nearest hundredth of a second, the times in the first two seconds when the voltage is equal to volts. () Let u5pt. Solve the equation 0 cos u5 in the interval 0 u,p. () Use the formula u5pt and your answers to part () to find t when 0 u,p and the voltage is equal to volts. 30. A water balloon leaves the air cannon at an angle of u with the ground and an initial velocity of 40 feet per second. The water balloon lands 30 feet from the cannon. The distance d traveled by θ the water balloon is given by the 30 ft formula d 5 3 v sin u where v is the initial velocity.

9 56 Trigonometric Equations a. Let x 5 u. Solve the equation (40) sin x to the nearest tenth of a degree. b. Use the formula x 5 u and your answer to part a to find the measure of the angle that the cannon makes with the ground. 3. It is important to understand the underlying mathematics before using the calculator to solve trigonometric equations. For example, Adrian tried to use the intersect feature of his graphing calculator to find the solutions of the equation cot u5sin A u p B in the interval 0 u p but got an error message. Follow the steps that Adrian used to solve the equation: () Enter Y 5 tan X and Y 5 sin A X p B into the Y menu. () Use the following viewing window to graph the equations: p Xmin 5 0, Xmax 5p, Xscl 5 6, Ymin 55, Ymax 5 5 (3) The curves seem to intersect at A p, 0 B. Press nd CALC 5 ENTER ENTER to select both curves. When the calculator asks for a guess, move the cursor near the intersection point using the arrow keys and then press ENTER. a. Why does the calculator return an error message? p b. Is u5 a solution to the equation? Explain. 3- USING FACTORING TO SOLVE TRIGONOMETRIC EQUATIONS We know that the equation 3x 5x 5 0 can be solved by factoring the left side and setting each factor equal to 0. The equation 3 tan u5 tan u 5 0 can be solved for tan u in a similar way. 3x 5x tan u5 tan u 5 0 (3x )(x ) 5 0 (3 tan u)(tan u) 5 0 3x 5 0 x tan u 5 0 tan u 5 0 3x 5 x 5 3 tan u5 tan u5 x 5 3 tan u5 3 In the solution of the algebraic equation, the solution is complete. The solution set is U 3, V. In the solution of the trigonometric equation, we must now find the values of u. There are two values of u in the interval 0 u,360 for which tan u5, one in the first quadrant and one in the third quadrant. The calculator will display the measure of the first-quadrant angle, which is also the reference angle for the third-quadrant angle.

10 Using Factoring to Solve Trigonometric Equations 57 ENTER: nd TAN ) ENTER DISPLAY: tan - () To the nearest tenth of a degree, the measure of u in the first quadrant is This is also the reference angle for the third-quadrant angle. In quadrant I: u In quadrant III: u 5 80 R There are two values of u in the interval 0 u,360 for which tan u5 3, one in the second quadrant and one in the fourth quadrant. ENTER: nd TAN 3 ) DISPLAY: ENTER tan - ( - /3) The calculator will display the measure of a fourth-quadrant angle, which is negative. To the nearest tenth of a degree, one measure of u in the fourth quadrant is 8.4. The opposite of this measure, 8.4, is the measure of the reference angle for the second- and fourth-quadrant angles. In quadrant II: u In quadrant IV: u The solution set of 3 tan u5 tan u 5 0 is {63.4, 6.6, 43.4, 34.6 } when 0 u,360. EXAMPLE 3 Find all values of u in the interval 0 u,p for which sin u 5 sin u. Solution Answer u5 3p How to Proceed () Multiply both sides of the equation by sin u: () Write an equivalent equation with 0 as the right side: (3) Factor the left side: (4) Set each factor equal to 0 and solve for sin u: (5) Find all possible values of u: 3 sin u 5 sin u sin usin u53 sin usin u3 5 0 ( sin u3)(sin u) 5 0 sin u3 5 0 sin u 5 0 sin u53 sin u5 3 sin u5 There is no value of u such that 3p sin u.. For sin u5, u5.

11 58 Trigonometric Equations EXAMPLE Find the solution set of 4 sin A 5 0 for the degree measures of A in the interval 0 A, 360. Solution METHOD METHOD Factor the left side. 4 sin A 5 0 ( sin A )( sin A ) 5 0 Solve for sin A and take the square root of each side of the equation. 4 sin A 5 0 sin A 5 0 sin A sin A 5 sin A 5 sin A 5 sin A 5 4 sin A 5 sin A 5 sin A 56 If sin A 5, A 5 30 or A If sin A 5, A 5 0 or A Answer {30, 50, 0, 330 } Note: The graphing calculator does not use the notation sin A, so we must enter the square of the trig function as (sin A) or enter sin (A). For example, to check the solution A 5 30 for Example : ENTER: 4 ( SIN 30 ) ) x ENTER 4 SIN 30 ) x ENTER DISPLAY: 4(sin(30)) - 4sin(30) Factoring Equations with Two Trigonometric Functions To solve an equation such as sin u cos u sin u 50, it is convenient to rewrite the left side so that we can solve the equation with just one trigonometric function value. In this equation, we can rewrite the left side as the product of two factors. Each factor contains one function.

12 Using Factoring to Solve Trigonometric Equations 59 sin u cos usin u50 sin u ( cos u) 5 0 sin u50 cos u 5 0 u 5 0 cos u5 u 5 80 cos u5 Since cos 60 5, R In quadrant II: u In quadrant III: u The solution set is {0, 0, 80, 40 }. EXAMPLE 3 Find, in radians, all values of u in the interval 0 u p that are in the solution set of: sec u csc u! csc u 50 Solution Factor the left side of the equation and set each factor equal to 0. sec u csc u! csc u50 csc u Asec u!b50 csc u50 sec u! 50 No solution sec u5! cos u5! 5! p p Since cos 4 5!, R 5 4. p 3p In quadrant II: u5p p 5p In quadrant III: u5p Answer U 3p 4, 5p 4 V Exercises Writing About Mathematics. Can the equation tan usin u tan u5 be solved by factoring the left side of the equation? Explain why or why not.. Can the equation (sin u)(cos u) sin u cos u 5 0 be solved by factoring the left side of the equation? Explain why or why not.

13 530 Trigonometric Equations Developing Skills In 3 8, find the exact solution set of each equation if 0 u, sin usin u tan u5 5. tan u sin u cos u5 cos u sin u cos ucos u50 In 9 4, find, to the nearest tenth of a degree, the values of u in the interval 0 u,360 that satisfy each equation. 9. tan u3 tan u cos u4 cos u sin u9 sin u cos u tan u4 tan u sec u7 sec u 5 0 In 5 0, find, to the nearest hundredth of a radian, the values of u in the interval 0 u,p that satisfy the equation. 5. tan u5 tan u cos u3 cos u sin u sin u sin u7 sin u csc u6 csc u cot u3 cot u Find the smallest positive value of u such that 4 sin u Find, to the nearest hundredth of a radian, the value of u such that sec u5sec u and p,u,p. 3. Find two values of A such that (sin A)(csc A) 5sin A. 3-3 USING THE QUADRATIC FORMULA TO SOLVE TRIGONOMETRIC EQUATIONS Not all quadratic equations can be solved by factoring. It is often useful or necessary to use the quadratic formula to solve a second-degree trigonometric equation. The trigonometric equation cos u4 cos u 5 0 is similar in form to the algebraic equation x 4x 5 0. Both are quadratic equations that cannot be solved by factoring over the set of integers but can be solved by using the quadratic formula with a 5, b 54, and c 5.

14 Using the Quadratic Formula to Solve Trigonometric Equations 53 Algebraic equation: Trigonometric equation: x 4x 5 0 cos u4 cos u b 6 "b x 5 4ac b 6 "b cos u5 4ac a a (4) 6 "(4) x 5 4()() (4) 6 "(4) cos u5 4()() () () 4 6! !6 8 x 5 4 cos u !8 4 6!8 x 5 4 cos u5 4 6! 6! x 5 cos u5 There are no differences between the two solutions up to this point. However, for the algebraic equation, the solution is complete. There are two values of x that make the equation true: x 5 or x 5.!! For the trigonometric equation, there appear to be two values of cos u. Can we find values of u for each of these two values of cos u? CASE cos u 5! There is no value of u such that cos u.. CASE cos u 5 " There are values of u in the first quadrant and in the fourth quadrant such that cos u is a positive number less than. Use a calculator to approximate these values to the nearest degree. ENTER: nd COS ( DISPLAY: nd ) ) cos - ((- ())/) ) ENTER To the nearest degree, the value of u in the first quadrant is 73. This is also the value of the reference angle. Therefore, in the fourth quadrant, u or 87. In the interval 0 u,360, the solution set of cos u4 cos u 5 0 is: {73, 87 }

15 53 Trigonometric Equations EXAMPLE a. Use three different methods to solve tan u 5 0 for tan u. b. Find all possible values of u in the interval 0 u,p. Solution a. METHOD : FACTOR METHOD : SQUARE ROOT tan u 5 0 tan u 5 0 (tan u)(tan u) 5 0 tan u5 tan u 5 0 tan u 5 0 tan u56 tan u5 tan u5 METHOD 3: QUADRATIC FORMULA tan u 5 0 a 5, b 5 0, c "0 6!4 tan u5 4()() () b. When tan u5, u is in quadrant I or in quadrant III. p In quadrant I: if tan u5, u5 4 This is also the measure of the reference angle. p In quadrant III: u5p 4 5 5p 4 When tan u5, u is in quadrant II or in quadrant IV. One value of u is p. The reference angle is 4 p 4. In quadrant II: u5p p 4 5 3p 4 In quadrant IV: u5p p 4 5 7p 4 Answer U p 4, 3p 4, 5p 4, 7p 4 V EXAMPLE Find, to the nearest degree, all possible values of B such that: 3 sin B 3 sin B 5 0 Solution a 5 3, b 5 3, c "(3) 3 6! !33 sin B 5 4(3)() (3) CASE 3!33 Let sin B !33 Since 6 < 0.46 is a number between and, it is in the range of the sine function.

16 Using the Quadratic Formula to Solve Trigonometric Equations 533 ENTER: nd SIN ( 3 DISPLAY: nd 33 ) ) 6 ) ENTER sin - (( - 3+ (33)) /6) The sine function is positive in the first and second quadrants. In quadrant I: B 5 7 In quadrant II: B Calculator Solution CASE 3!33 Let sin B !33 Since 6 <.46 is not a number between and, it is not in the range of the sine function. There are no values of B such that 3!33 sin B 5 6. Enter Y 5 3 sin X 3 sin X into the Y menu. ENTER: Y 3 SIN X,T,,n ) x 3 SIN X,T,,n ) With the calculator set to DEGREE mode, graph the function in the following viewing window: Xmin 5 0, Xmax 5 360, Xscl 5 30, Ymin 55, Ymax 5 5 The solutions are the x-coordinates of the x- intercepts, that is, the roots of Y. Use the zero function of your graphing calculator to find the roots. Press nd CALC. Use the arrows to enter a left bound to the left of one of the zero values, a right bound to the right of the zero value, and a guess near the zero value. The calculator will display the coordinates of the point at which the graph intersects the x-axis. Repeat to find the other root. * Zero X=7.08 Y=0 * Zero X= Y=0 As before, we find that the solutions in the interval 0 u,360 are approximately 7 and 53. Answer B n or B n for integral values of n.

17 534 Trigonometric Equations Exercises Writing About Mathematics. The discriminant of the quadratic equation tan u4 tan u5 5 0 is 4. Explain why the solution set of this equation is the empty set.. Explain why the solution set of csc ucsc u50 is the empty set. Developing Skills In 3 4, use the quadratic formula to find, to the nearest degree, all values of u in the interval 0 u,360 that satisfy each equation sin u7 sin u tan u tan u cos u 5 5 cos u 6. 9 sin u6 sin u5 7. tan u3 tan u cos u7 cos u cot u3 cot u sec u sec u csc u csc u5. tan u (tan u) 5 3 sin u cos u 5 cos u 4. 5 sin u sin u 5. Find all radian values of u in the interval 0 u,p for which 5 sin u. 6. Find, to the nearest hundredth of a radian, all values of u in the interval 0 u,p for cos u which cos u. 3-4 USING SUBSTITUTION TO SOLVE TRIGONOMETRIC EQUATIONS INVOLVING MORE THAN ONE FUNCTION When an equation contains two different functions, it may be possible to factor in order to write two equations, each with a different function. We can also use identities to write an equivalent equation with one function. The equation cos usin u5 cannot be solved by factoring. We can use the identity cos usin u5 to change the equation to an equivalent equation in sin u by replacing cos u with sin u. cos usin u5 sin u sin u5 sin usin u50 sin u (sin u) 5 0 sin u50 sin u 5 0 u sin u u 5 80 u

18 Using Substitution to Solve Trigonometric Equations Involving More than One Function 535 Check u 5 0 Check u 5 80 Check u cos u sin u5 cos 08 sin 08 5? 0 5? cos u sin u5 cos 808 sin 808 5? () 0 5? In the interval 0 u,360, the solution set of cos usin u5 is {0, 90, 80 }. Is it possible to solve the equation cos usin u5 by writing an equivalent equation in terms of cos u? To do so we must use an identity to write sin u in terms of cos u. Since sin u5 cos u, sin u5 6" cos u. () Write the equation: cos usin u 5 () Replace sin u with cos u 6" cos u 5 6" cos u: cos u sin u5 cos 908 sin 908 5? 0 5? (3) Isolate the radical: 6" cos u 5 cos u (4) Square both sides of cos u5 cos ucos 4 u the equation: (5) Write an equivalent equation with the right cos ucos 4 u50 side equal to 0: (6) Factor the left side: cos u ( cos u) 5 0 (7) Set each factor equal to 0: cos u50 cos u50 cos u50 5 cos u u cos u u 5 70 u u This approach uses more steps than the first. In addition, because it involves squaring both sides of the equation, an extraneous root, 70, has been introduced. Note that 70 is a root of the equation cos ucos 4 u50 but is not a root of the given equation. cos usin u5 cos? 70 sin 70 5 (0) () 5? 0

19 536 Trigonometric Equations Any of the eight basic identities or the related identities can be substituted in a given equation. EXAMPLE Find all values of A in the interval 0 A, 360 such that sin A 5 csc A. Solution How to Proceed () Write the equation: sin A 5 csc A () Replace csc A with sin A: sin A 5 sin A (3) Multiply both sides of the sin A sin A 5 equation by sin A: (4) Write an equivalent equation sin A sin A 5 0 with 0 as the right side: (5) Factor the left side: ( sin A )(sin A ) 5 0 (6) Set each factor equal to 0 sin A 5 0 sin A 5 0 and solve for sin A: sin A 5 sin A 5 sin A 5 A 5 70 A 5 30 or A 5 50 Answer A 5 30 or A 5 50 or A 5 70 Often, more than one substitution is necessary to solve an equation. EXAMPLE If 0 u,p, find the solution set of the equation sin u 53 cot u. Solution How to Proceed () Write the equation: sin u53 cot u cos u () Replace cot u with sin u: sin u53a sin u u B (3) Multiply both sides of the sin u53 cos u equation by sin u: (4) Replace sin u with cos u: ( cos u) 5 3 cos u (5) Write an equivalent equation cos u53 cos u in standard form: cos u3 cos u50 cos u3 cos u 5 0

20 Using Substitution to Solve Trigonometric Equations Involving More than One Function 537 (6) Factor and solve for cos u: ( cos u)(cos u) 5 0 (7) Find all values of u in the cos u 5 0 cos u 5 0 given interval: cos u5 cos u5 cos u5 p u5 3 5p or u5 3 No solution Answer U p 3, 5p 3 V The following identities from Chapter 0 will be useful in solving trigonometric equations: Reciprocal Identities Quotient Identities Pythagorean Identities sec u5cos u sin u tan u5cos u cos usin u5 csc u5sin u cos u cot u5sin u tan u5sec u cot u5tan u cot u 5 csc u Exercises Writing About Mathematics. Sasha said that sin u cos u 5 has no solution. Do you agree with Sasha? Explain why or why not.. For what values of u is sin u5" cos u true? Developing Skills In 3 4, find the exact values of u in the interval 0 u,360 that satisfy each equation. 3. cos u3 sin u cos u4 sin u csc ucot u sin u 5 csc u 7. sin u3 cos u tan u5cot u 9. cos u5sec u 0. sin u5csc u. tan u5cot u. cos u5sin u 3. cot u5csc u 4. sin utan u cot u50

21 538 Trigonometric Equations Applying Skills 5. An engineer would like to model a piece for a factory machine on his computer. As shown in the figure, the machine consists of a link fixed to a circle at point A. The other end of the link is fixed to a slider at point B. As the circle rotates, point B slides back and forth between the two ends of the slider (C and D). The movement is restricted so that u, the measure of AOD, is in the interval 45 u 45. The motion of point B can be described mathematically by the formula O u A C B D CB 5 r(cos u) "l r sin u where r is the radius of the circle and l is the length of the link. Both the radius of the circle and the length of the link are inches. a. Find the exact value of CB when: () u 5 30 () u545. b. Find the exact value(s) of u when CB 5 inches. c. Find, to the nearest hundredth of a degree, the value(s) of u when CB 5.5 inches. 3-5 USING SUBSTITUTION TO SOLVE TRIGONOMETRIC EQUATIONS INVOLVING DIFFERENT ANGLE MEASURES If an equation contains function values of two different but related angle measures, we can use identities to write an equivalent equation in terms of just one variable. For example: Find the value(s) of u such that sin u sin u50. Recall that sin u 5 sin u cos u. We can use this identity to write the equation in terms of just one variable, u, and then use any convenient method to solve the equation. () Write the equation: sin u sin u50 () For sin u, substitute its equal, sin u cos u sin u50 sin u cos u: (3) Factor the left side: sin u ( cos u) 5 0 (4) Set each factor equal to zero sin u50 cos u 5u and solve for u: u50 cos u5 or u580 cos u5 u560 or u5300

22 Using Substitution to Solve Trigonometric Equations Involving Different Angle Measures 539 The degree measures 0,60, 80, and 300 are all of the values in the interval 0 u,360 that make the equation true.any values that differ from these values by a multiple of 360 will also make the equation true. EXAMPLE Find, to the nearest degree, the roots of cos u cos u 50. Solution How to Proceed () Write the given equation: () Use an identity to write cos u in terms of cos u: (3) Write the equation in standard form: (4) The equation cannot be factored over the set of integers. Use the quadratic formula: (5) When we use a calculator to approximate the value of!3 arccos, the calculator will return the value. Cosine is negative in both the second and the third quadrants. Therefore, there is both a second-quadrant and a third-quadrant angle such!3 that cos u5 : (6) When we use a calculator to approximate the value of!3 arccos, the calculator will return an error message because!3. is not in the domain of arccosine: cos u cos u50 cos u cos u50 cos u cos u 5 0 () 6 "() cos u5 4()() () 6! !3 cos-((- (3))/) u 5 R u ERR:DOMAIN : Quit :Goto Answer To the nearest degree, u5 or u 549.

23 540 Trigonometric Equations EXAMPLE Find, to the nearest degree, the values of u in the interval 0 u 360 that are solutions of the equation sin (90 u) cos u 5. Solution Use the identity sin (90 u) 5 cos u. sin(90 u) cos u5 cos u cos u5 3 cos u5 cos u53 To the nearest degree, a calculator returns the value of u as 48. In quadrant I, u548 and in quadrant IV, u Answer u548 or u 53 The basic trigonometric identities along with the cofunction, double-angle, and half-angle identities will be useful in solving trigonometric equations: Cofunction Identities Double-Angle Identities Half-Angle Identities cos u5sin (90 u) sin (u) 5 sin u cos u sin u 5 sin u5cos (90 u) cos (u) 5 cos usin u cos u 5 tan u5cot (90 u) tan u tan (u) 5 tan tan u 5 u cot u5tan (90 u) sec u5csc (90 u) csc u5sec (90 u) 6# cos u 6# cos u 6# cos u cos u Note that in radians, the right sides of the cofunction identities are written in terms of p u. Exercises Writing About Mathematics. Isaiah said that if the equation cos x cos x 5 is divided by, an equivalent equation is cos x cos x 5. Do you agree with Isaiah? Explain why or why not.. Aaron solved the equation sin u cos u5cos u by first dividing both sides of the equation by cos u. Aaron said that for 0 u p, the solution set is U p 6, 5p 6 V. Do you agree with Aaron? Explain why or why not.

24 44C3.pgs 8//08 :56 PM Page Using Substitution to Solve Trigonometric Equations Involving Different Angle Measures Developing Skills In 3 0, find the exact values of u in the interval 0 u 360 that make each equation true. 3. sin u cos u cos u sin u 5 5. sin u sin u tan u 5 cot u 7. cos u cos u sin u cos u sin u 5 cos u 0. 3 cos u 4 cos u 5 0 In 8, find all radian measures of u in the interval 0 u p that make each equation true. Express your answers in terms of p when possible; otherwise, to the nearest hundredth.. cos u 5 cos u cos u. sin u sin u sin u 4 sin u cos u cos u 5 3 sin u 5. 3 sin u 5 tan u 6. sin (90 u) cos u cos u 3 sin u cos u 5 8. ( sin u cos u) 4 sin u 5 0 Applying Skills 9. Martha swims 90 meters from point A on the north bank of a stream to point B on the opposite bank. Then she makes a right angle turn and swims 60 meters from point B to point C, another point on the north bank. If m CAB 5 u, then m ACB 5 90 u. C A u 90 u d a. Let d be the width of the stream, the length of the perpendicular distance from B to AC. Express d in terms of sin u. B b. Express d in terms of sin (90 u). c. Use the answers to a and b to write an equation. Solve the equation for u. d. Find d, the width of the stream. 0. A pole is braced by two wires of equal length as shown in the diagram. One wire, AB, makes an angle of u with the ground, and the other wire, CD, makes an angle of u with the ground. If FD 5.75FB, find, to the nearest degree, the measure of u: D B a. Let AB = CD = x, FB 5 y, and FD 5.75y. Express sin u and sin u in terms of x and y. b. Write an equation that expresses a relationship between sin u and sin u and solve for u to the nearest degree. u C u F A

25 54 Trigonometric Equations CHAPTER SUMMARY A trigonometric equation is an equation whose variable is expressed in terms of a trigonometric function value. The following table compares the degree measures of u from 90 to p 360, the radian measures of u from to p, and the measure of its reference angle. Fourth First Second Third Fourth Quadrant Quadrant Quadrant Quadrant Quadrant Angle 90,u,0 0,u,90 90,u,80 80,u,70 70,u,360 p 3p 3p p,u,0 0,u, p,u,u p,u,,u,p Reference u u 80 u u u Angle u u p u u p p u To solve a trigonometric equation:. If the equation involves more than one variable, use identities to write the equation in terms of one variable.. If the equation involves more than one trigonometric function of the same variable, separate the functions by factoring or use identities to write the equation in terms of one function of one variable. 3. Solve the equation for the function value of the variable. Use factoring or the quadratic formula to solve a second-degree equation. 4. Use a calculator or your knowledge of the exact function values to write one value of the variable to an acceptable degree of accuracy. 5. If the measure of the angle found in step 4 is not that of a quadrantal angle, find the measure of its reference angle. 6. Use the measure of the reference angle to find the degree measures of each solution in the interval 0 u,360 or the radian measures of each solution in the interval 0 u,p. 7. Add 360n (n an integer) to the solutions in degrees found in steps 4 and 6 to write all possible solutions in degrees. Add pn (n an integer) to the solutions in radians found in steps and 4 to write all possible solutions in radians. VOCABULARY 3- Trigonometric equation

26 Review Exercises 543 REVIEW EXERCISES In 0, find the exact values of x in the interval 0 x 360 that make each equation true.. cos x 5 0.!3 sin x 5 sin x! 3. sec x 5 sec x 4. cos x cos x cos x sin x sin x tan x 3 cot x cos x sec x sin x cos x tan x 5 tan x 0. cos 3 3 x 4 cos x 5 0 In, find, to the nearest hundredth, all values of u in the interval 0 u p that make each equation true.. 7 sin u3 5. 5(cos u) 5 6 cos u 3. 4 sin u3 sin u cos ucos u tan u4 tan u sec u0 sec u tan u 54 tan u 8. sin u cos u50 sin u 9. cos u cos u cos u 5 0. tan u6 tan u50. cos u cos ucos u Explain why the solution set of tan usec u50 is the empty set. 4. In ABC,m A 5uand m B 5 u.the C altitude from C intersects AB at D and AD : DB 5 5 :. a. Write tan u and tan u as ratios of the sides of ADC and BDC, respectively. A D u b. Solve the equation for tan u found in a for CD. u B c. Substitute the value of CD found in b into the equation of tan u found in a. d. Solve for u. e. Find the measures of the angles of the triangle to the nearest degree.

27 544 Trigonometric Equations Exploration For () (6): a. Use your knowledge of geometry and trigonometry to express the area, A, of the shaded region in terms of u. b. Find the measure of u when A square unit or explain why there is no possible value of u. Give the exact value when possible; otherwise, to the nearest hundredth of a radian. () () u u p p 0, u, 0, u, (3) (4) u O u p 0, u, 0, u,p (5) (6) u O u O p,u#p 0,u, p

28 Cumulative Review 545 CUMULATIVE REVIEW CHAPTERS 3 Part I Answer all questions in this part. Each correct answer will receive credits. No partial credit will be allowed.. The expression (5) 0 3(7) 3 is equal to () 7 ()!3 9 (3) 3 (4) 9. The sum of 4 a a and a a is () 4 a a 5 (3) 4 a 3 a () a 3 7 (4) 4 a a 4 3. The fraction is equivalent to!5 ()!5 (3) +!5 ()!5 (4)!5 4. The complex number i i 0 can be written as () 0 () (3) (4) i 4 5. a S() n n T is equal to n5 () 5 () (3) (4) 5 6. When the roots of a quadratic equation are real and irrational, the discriminant must be () zero. () a positive number that is a perfect square. (3) a positive number that is not a perfect square. (4) a negative number. 7. When f(x) 5 x and g(x) 5 x, then g(f(x)) equals () 4x (3) 4x () x x (4) x 8. If log x 5 log a 3 log b, then x equals a () a b (3) 3 3 b () a a! 3 b (4)! 3 b 9. What number must be added to the binomial x 5x in order to change it into a trinomial that is a perfect square? () () 4 (3) (4) 5

29 546 Trigonometric Equations 0. If f(x) is a function from the set of real numbers to the set of real numbers, which of the following functions is one-to-one and onto? () f(x) 5 x (3) f(x) 5 x () f(x) = x (4) f(x) 5 x x Part II Answer all questions in this part. Each correct answer will receive credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only credit.. Sketch the graph of the inequality y x x 3.. When f(x) 5 4x, find f (x), the inverse of f(x). Part III Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only credit. 3. Write the first six terms of the geometric function whose first term is and whose fourth term is The endpoints of a diameter of a circle are (, 5) and (4, ). Write an equation of the circle. Part IV Answer all questions in this part. Each correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only credit. 5. If csc u53, and cos u,0, find sin u, cos u, tan u, cot u and sec u Solve for x and write the roots in a + bi form: x 5 x 5.