Solve Trigonometric Equations. Solve a trigonometric equation

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1 14.4 a.5, a.6, A..A; P.3.D TEKS Before Now Solve Trigonometric Equations You verified trigonometric identities. You will solve trigonometric equations. Why? So you can solve surface area problems, as in Ex. 43. Key Vocabulary extraneous solution, p. 5 In Lesson 14.3, you verified trigonometric identities. In this lesson, you will solve trigonometric equations. To see the difference, consider the following: sin x 1 cos x 5 1 Equation 1 sin x 5 1 Equation Equation 1 is an identity because it is true for all real values of x. Equation, however, is true only for some values of x. When you find these values, you are solving the equation. E XAMPLE 1 Solve a trigonometric equation Solve sin x Ï } Solution First isolate sin x on one side of the equation. sin x Ï } Write original equation. sinx 5 Ï } 3 Add Ï } 3 to each side. sin x 5 Ï} 3 }} Divide each side by. One solution of sin x 5 Ï} 3 }} in the interval 0 x < π is x 5 sin 1 Ï} 3 }} 5 } p. 3 The other solution in the interval is x 5 π } p 5 }} p. Moreover, because 3 3 y 5 sin x is periodic, there will be infinitely many solutions. WRITE GENERAL SOLUTION To write the general solution of a trigonometric equation, you can add multiples of the period to all the solutions from one cycle. You can use the two solutions found above to write the general solution: x 5 } p 1 np or x 5 }} p 1 np (where n is any integer) 3 3 CHECK You can check the answer by graphing y 5 sin x and y 5 Ï} 3 }} in the same coordinate plane. Then find the points where the graphs intersect. You can see that there are infinitely many such points. 1 y y 5 3 π y 5 sin x x 14.4 Solve Trigonometric Equations 931

2 E XAMPLE Solve a trigonometric equation in an interval Solve 9 tan x in the interval 0 x < p. 9 tan x Write original equation. 9tan x 5 1 Subtract from each side. tan x 5 1 } 9 Divide each side by 9. REVIEW INVERSE FUNCTIONS For help with inverse trigonometric functions, see p tan x 56} 1 Take square roots of each side. 3 Using a calculator, you find that tan 1 1 } 3 ø 0.3 and tan } 3 ø 0.3. Therefore, the general solution of the equation is: x ø nπ or x ø nπ (where n is any integer) c The specific solutions in the interval 0 x < π are: xø 0.3 x ø π ø.80 xø π ø x ø π ø E XAMPLE 3 Solve a real-life trigonometric equation OCEANOGRAPHY The water depth d for the Bay of Fundy can be modeled by d cos }} p t 6. where d is measured in feet and t is the time in hours. If t 5 0 represents midnight, at what time(s) is the water depth 7 feet? High tide Low tide ANOTHER WAY For alternative methods for solving the problem in Example 3, turn to page 938 for the Problem Solving Workshop. Solution Substitute 7 for d in the model and solve for t cos }} p t 5 7 Substitute 7 for d cos p }} 6. t 58 Subtract 35 from each side. cos p }} 6. t 5 1 Divide each side by 8. p }} 6. t 5 nπ cos u 5 1 when u 5 np. t 5 1.4n Solve for t. c On the interval 0 t 4 (representing one full day), the water depth is 7 feet when t 5 1.4(0) 5 0 (that is, at midnight) and when t 5 1.4(1) (that is, at 1:4 P.M.). 93 Chapter 14 Trigonometric Graphs, Identities, and Equations

3 GUIDED PRACTICE for Examples 1,, and 3 1. Find the general solution of the equation sin x Solve the equation 3 csc x 5 4 in the interval 0 x < π. 3. OCEANOGRAPHY In Example 3, at what time(s) is the water depth 63 feet? E XAMPLE 4 TAKS PRACTICE: Multiple Choice What is the general solution of sin 3 x 9 sin x 5 0? A x 5 } p 1 nπ or x 5 }} 3p 1 nπ B x 5 } p 1 nπ or x 5 π 1 nπ C x 5 π 1 nπ D x 5 nπ or x 5 π 1 nπ Solution sin 3 x 9 sin x 5 0 Write original equation. sin x (sin x 9) 5 0 Factor out sin x. ELIMINATE SOLUTIONS Because sin x is never less than 1 or greater than 1, there are no solutions of sin x 53 and sin x 5 3. sin x (sin x 1 3)(sin x 3) 5 0 Factor difference of squares. Set each factor equal to 0 and solve for x, if possible. sin x 5 0 sin x sin x x 5 0 or x 5 π sin x 53 sin x 5 3 The only solutions in the interval 0 x < π are x 5 0 and x 5 π. The general solution is x 5 nπ or x 5 π 1 nπ where n is any integer. c The correct answer is D. A B C D E XAMPLE 5 Use the quadratic formula Solve cos x 5 cos x in the interval 0 x p. Solution Because the equation is in the form au 1 bu 1 c 5 0 where u 5 cos x, you can use the quadratic formula to solve for cos x. cos x 5 cos x (5) 6 Ï}} (5) cos x 5 4(1)() }}}}}}}}}}}} (1) Ï} 17 }}}} ø 4.56 or 0.44 Write original equation. Quadratic formula Simplify. Use a calculator. x 5 cos x 5 cos Use inverse cosine. No solution ø 1.1 Use a calculator, if possible. c In the interval 0 x π, the only solution is x ø Solve Trigonometric Equations 933

4 EXTRANEOUS SOLUTIONS When solving a trigonometric equation, it is possible to obtain extraneous solutions. So, you should always check your solutions in the original equation. E XAMPLE 6 Solve an equation with an extraneous solution Solve 1 1 cos x 5 sin x in the interval 0 x < p. 11 cos x 5 sin x Write original equation. REVIEW FOIL METHOD For help multiplying binomials, see p. 45. (1 1 cos x) 5 (sin x) Square both sides. 1 1 cos x 1 cos x 5 sin x Multiply. 1 1 cos x 1 cos x 5 1 cos x Pythagorean identity cos x 1 cos x 5 0 Quadratic form cosx (cos x 1 1) 5 0 Factor out cos x. cos x 5 0 or cos x Zero product property cos x 5 0 or cos x 51 Solve for cos x. On the interval 0 x < π, cos x 5 0 has two solutions: x 5 } p or x 5 }} 3p. On the interval 0 x < π, cos x 51 has one solution: x 5 π. Therefore, 1 1 cos x 5 sin x has three possible solutions: x 5 p }, π, and 3p }}. CHECK To check the solutions, substitute them into the original equation and simplify. 11 cos x 5 sin x 11 cos x 5 sin x 11 cos x 5 sin x 11 cos } p 0 sin } p 11 cos p 0 sin p 11 cos }} 3p 0 sin }} 3p (1) Þ1 1 y y cos x c The apparent solution x 5 }} 3p is extraneous because it does not check in the original equation. The only solutions in the interval 0 x < π are x 5 } p and x 5 π. Graphs of each side of the original equation confirm the solutions. π y 5 sin x x GUIDED PRACTICE for Examples 4, 5, and 6 Find the general solution of the equation. 4. sin 3 x sin x cos x 5 Ï } 3 sin x Solve the equation in the interval 0 x p. 6. sin x 5 csc x 7. tan x sin x tan x Chapter 14 Trigonometric Graphs, Identities, and Equations

5 14.4 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS1 for Exs. 5, 13, and 43 5 TAKS PRACTICE AND REASONING Exs. 15, 36, 4, 44, 46, and 47 5 MULTIPLE REPRESENTATIONS Ex VOCABULARY What is the difference between a trigonometric equation and a trigonometric identity?. WRITING Describe several techniques for solving trigonometric equations. EXAMPLE 1 on p. 931 for Exs CHECKING SOLUTIONS Verify that the given x-value is a solution of the equation cos x 5 5 0, x 5 4π 4. π sec x 1 π 5 0, x 5 π 5. 1 sin x 3 5 0, x 5 p } tan 3 x 5 5 0, x 5 p } 4 7. cos 4 x cos x 5 0, x 5 p } 8. 3 cot 4 x cot x 4 5 0, x 5 7p }} 6 GENERAL SOLUTIONS Find the general solution of the equation. 9. sin x Ï } 3 csc x tanx Ï } sin x 1 Ï } 5sin x cos x tan x TAKS REASONING What is the general solution of the equation 4 sin x 5 sin x 1 1? A C x 5 } p 1 nπ or x 5 }} 7p 1 nπ B x 5 } p 1 nπ or x 5 }} 5p 1 nπ x 5 } p 1 nπ or x 5 }} 5p 1 nπ D x 5 } p 1 nπ or x 5 }} 7p 1 nπ EXAMPLE on p. 93 for Exs SOLVING EQUATIONS Solve the equation in the interval 0 x < p sin x tan x Ï } cosx 5 cos x sin x tan x cos x ERROR ANALYSIS Describe and correct the error in solving the equation in the interval 0 x } p.. sin x 5 1 } sin x 3. cos x 51 sin x 5 1 } cos x 5 1 } x 5 p } 6 x 5 p }} 3 EXAMPLE 4 on p. 933 for Exs. 4 9 GENERAL SOLUTIONS Find the general solution of the equation. 4. sin x cos x 3 cos x Ï } 3 cos x tan x cos x sin 3 x 5 sin x 7. tan 4 x tan x Ï } cos x 5 cos x cos x 5 Ï } 3 sin x 14.4 Solve Trigonometric Equations 935

6 EXAMPLES 5 and 6 on pp for Exs SOLVING Solve the equation in the given interval. Check your solutions. 30. sec x csc x 5 sec x; 0 x < π 31. Ï } 3 cos x 5 cos x tan x; 0 x π 3. sin x cos x 1 5 0; 0 x < π 33. sin x 1 5 sin x 3 5 0; p } x < p } 34. tan x 3 tan x 1 5 0; 0 x π 35. cos x 1 sin x tan x 5 ; π x < π 36. TAKS REASONING What are the points of intersection of the graphs of y 5 4 sin x 1 1 and y 5 sin x 1 on the interval 0 x < π? A 1 p } 6, 3, 1 p }, 3 B 1 p } 6, 3, 1 5p }} 6, 3 C 1} p, 3, 1}} 7p, 6 3 D 1} p, 6 3, 1}} 11p 6,3 INTERSECTION POINTS Find the points of intersection of the graphs of the given functions in the interval 0 x < p. 37. y 5 cos x 38. y 5 9 sin x 39. y 5 Ï } 3 tan x y 5 cos x 1 y 5 sin x 1 8 sin x y 5 Ï } 3 tan x 40. CHALLENGE A number c is a fixed point of a function f if f(c) 5 c. For example, 0 is a fixed point of f(x) 5 sin x because f(0) 5 sin a. Reasoning Use graphs to explain why the function g(x) 5 cos x has only one fixed point. b. Graphing Calculator Find the fixed point of g(x) 5 cos x. PROBLEM SOLVING EXAMPLE 3 on p. 93 for Exs WIND SPEED The average wind speed s (in miles per hour) in the Boston Harbor can be approximated by s sin }} p (t 1 3) where t is the 180 time in days, with t 5 0 representing January 1. On which days of the year is the average wind speed 10 miles per hour? 4. TAKS REASONING The number of degrees u north of due east (u > 0) or south of due east (u < 0) that the sun rises in Cheyenne, Wyoming, can be modeled by u(t) 5 31 sin 1}} p t where t is the time in days, with t 5 1 representing January 1. Use an algebraic method to find at what day(s) the sun is 08 north of due east at sunrise. Explain how you can use the graph of u(t) to check your answer WORKED-OUT SOLUTIONS on p. WS1 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS

7 43. MULTIPLE REPRESENTATIONS The surface area S of a honeycomb cell can be estimated by the equation shown at the right. In the equation, h is the height (in inches), s is the width of a side (in inches), and u is the angle (in degrees) indicated in the diagram. a. Using a Diagram Use the values of h and s in the diagram to simplify the equation. b. Making a Table Use a graphing calculator to make a table for the function from part (a). For what value(s) of u does S 5 9 square inches? c. Drawing a Graph Use a graphing calculator to graph the function from part (a). What value of u minimizes the surface area? u s h S 5 6hs 1 3 } s 1 Ï} 3 cos u }}}}} sin u 44. TAKS REASONING The power P (in watts) used by a microwave oven is the product of the voltage V (in volts) and the current I (in amperes). Suppose the voltage and current can be modeled by where t is the time (in seconds). V5 170 cos 10πt and I cos 10πt a. Model Write the function P(t) for the power used by the microwave. b. Solve At what times does the microwave use 375 watts of power? c. Graphing Calculator Graph the function P(t). Describe how the graph differs from that of a cosine function of the form y 5 a cos bt. 45. CHALLENGE Matrix multiplication can be used to rotate a point (x, y) counterclockwise about the origin through an angle u. The coordinates of the resulting point (x9, y9) are determined by the matrix equation shown at the right. a. The point (, 3) is rotated counterclockwise about the origin through an angle of } p. What are the 3 coordinates of the resulting point? b. Through what angle u must the point (6, ) be rotated to produce (x9, y9) 5 (3Ï } 3 1, Ï } 3 1 3)? (x9, y9) Fcos u sin u sin u cos ugfyg 5Fy9G x x x9 y u (x, y) MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson.4; TAKS Workbook REVIEW Lesson.3; TAKS Workbook 46. TAKS PRACTICE The speed of a falling object increases 3 feet per second each second it falls. From a high cliff, Andrew throws an object downward with an initial speed of 8 feet per second. Which equation represents the speed s (in feet per second) of the falling object after t seconds? TAKS Obj. 1 A s 53t 1 8 B s 5 3t 1 8 C s 5 8t 1 3 D s 5 3t 47. TAKS PRACTICE What are the coordinates of the y-intercept of the graph of 3x 1 4y 5 4? TAKS Obj. 3 F (8, 0) G 1 0, 4 } 3 H (0, 6) J (0, 8) EXTRA PRACTICE for Lesson 14.4, p. 103 ONLIN E QUIZ a t classzone.com 937

8 LESSON 14.4 TEKS a.5, a.6, A..A; P.3.D Using ALTERNATIVE METHODS Another Way to Solve Example 3, page 93 MULTIPLE REPRESENTATIONS In Example 3 on page 93, you solved a trigonometric equation algebraically. You can also solve a trigonometric equation using a table or using a graph. P ROBLEM OCEANOGRAPHY The water depth d for the Bay of Fundy can be modeled by d cos }} p t 6. where d is measured in feet and t is the time in hours. If t 5 0 represents midnight, at what time(s) is the water depth 7 feet? M ETHOD 1 Using a Table The problem requires solving the equation 35 8 cos p }} 6. t 5 7. One way to solve this equation is to make a table of values. You can use a graphing calculator to make the table. STEP 1 Enter the function y cos }} p x 6. into a graphing calculator. Note that time is now represented by x and water depth is now represented by y. Y1=35-8cos( X/6.) Y= Y3= Y4= Y5= Y6= Y7= STEP Make a table of values for the function. Set the table so that the x-values start at 0 and increase in increments of 0.1. (Be sure that the calculator is set in radian mode.) X X=0 Y STEP 3 Scroll through the table to find all the times x at which the water depth y is 7 feet. On the interval 0 x 4 (which represents one full day), you can see that the function equals 7 when x is 0 and 1.4. X Y X=1.4 c The water depth is 7 feet when x 5 0 (that is, at midnight) and when x (that is, at 1:4 P.M.). 938 Chapter 14 Trigonometric Graphs, Identities, and Equations

9 M ETHOD Using a Graph Another approach is to use a graph to solve the equation 35 8 cos }} p t 5 7. You can use a graphing calculator to make the graph. 6. STEP 1 Enter the functions y cos }} p x 6. and y 5 7 into a graphing calculator. Again, note that time is now represented by x and water depth is now represented by y. Y1=35-8cos( X/6.) Y=7 Y3= Y4= Y5= Y6= Y7= STEP Graph the functions. Set your calculator in radian mode. Adjust the viewing window so that you can see where the graphs intersect on the interval 0 x 4. STEP 3 Find the intersection points of the two graphs using the intersect feature of the graphing calculator. On the interval 0 x 4, the graphs intersect at (0, 7) and (1.4, 7). Because x represents the number of hours since midnight, you know that the water depth is 7 feet at midnight and 1:4 P.M. Intersection X=1.4 Y=7 P RACTICE SOLVING EQUATIONS Solve the equation using a table and using a graph sin p } 4 x cos p } 6 x cos πx sin p } x sin p }} 0 x cos } p 5 1 x }} p WHAT IF? In the problem on page 938, suppose you want to find the time(s) when the depth of the water in the Bay of Fundy is 15 feet. Find the time(s) using a table and using a graph. 8. WRITING Explain why the equation sin x has no solution. How does a graph show this? 9. BUOY An ocean buoy bobs up and down as waves travel past it. The buoy s displacement d (in feet) with respect to sea level can be modeled by d 5 3 sin πt where t is the time (in seconds). During the one second interval 0 t 1, when is the buoy 1.5 feet above sea level? Solve the problem using a table and using a graph. Using Alternative Methods 939

10 MIXED REVIEW FOR TEKS TAKS PRACTICE Lessons MULTIPLE CHOICE 1. AMUSEMENT PARK At an amusement park, you watch your friend go on a ride that simulates free-fall. You are standing 00 feet from the base of the ride as it slowly begins to pull your friend to the top. The ride is 10 feet tall. Which equation gives your friend s distance d (in feet) from the top of the ride as a function of the angle of elevation? TEKS a.4 d 10 d Your friend 4. RATE OF CHANGE In calculus, it can be shown that the rate of change of the function f(x) 5csc x sin x is given by this expression: csc x cot x cos x Which expression is equivalent to csc x cot x cos x? TEKS a.4 F cos x G cot x H cos x cot x J cos x csc x classzone.com GRIDDED ANSWER AMPLITUDE What is the amplitude of the graph shown? TEKS a.4 A d 5 50 }}} tan u You u 00 ft B d tan u π 1 y π π (π, 1) π x C d 5 50 tan u D d }}} tan u. BICYCLING You put a reflector on a spoke of your bicycle wheel. As you ride your bicycle, the reflector s height h (in inches) above the ground is modeled by h cos πt where t is the time (in seconds). What is the frequency of the function? TEKS a.4 F 1 G p } H π J π 3. HOT AIR BALLOON You stand 80 feet from the launch site of a hot air balloon traveling directly upward. What is the angle of elevation from you to the balloon when the balloon s height is 150 feet? TEKS a.4 A 8.18 B 3.8 C D (0, 7) 6. ICE CREAM PRODUCTION The number n (in millions) of gallons of ice cream produced in the United States can be approximated by n sin p 1}} t where t is the time (in days) with t 5 1 representing January 1. What value of t corresponds to the first day that 15 million gallons of ice cream will be produced? Round your answer to the nearest integer. TEKS a Chapter 14 Trigonometric Graphs, Identities, and Equations