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6-5 Trigonometric Equations with frequencies 64 and 72 hertz, respectively. If both sounds are emitted simultaneously, a beat frequency results. (A) Show that 0.5 cos 128t ⫺ 0.5 cos 144t ⫽ sin 8t sin 136t (The product form is more useful to sound engineers.) (B) Graph each equation in a different viewing window for 0 ⱕ t ⱕ 0.25: 485 —Beat Frequencies. y ⫽ 0.25 cos 256t and 48. Music— y ⫽ ⫺0.25 cos 288t are equations of sound waves with frequencies 128 and 144 hertz, respectively. If both sounds are emitted simultaneously, a beat frequency results. (A) Show that 0.25 cos 256t ⫺ 0.25 cos 288t ⫽ 0.5 sin 16t sin 272t (The product form is more useful to sound engineers.) (B) Graph each equation in a different viewing window for 0 ⱕ t ⱕ 0.125: y ⫽ 0.5 cos 128t y ⫽ 0.25 cos 256t y ⫽ ⫺0.5 cos 144t y ⫽ ⫺0.25 cos 288t y ⫽ 0.5 cos 128t ⫺ 0.5 cos 144t y ⫽ 0.25 cos 256t ⫺ 0.25 cos 288t y ⫽ sin 8t sin 136t y ⫽ 0.5 sin 16t sin 272t Section 6-5 Trigonometric Equations Solving Trigonometric Equations Using an Algebraic Approach Solving Trigonometric Equations Using a Graphing Utility The first four sections of this chapter consider trigonometric equations called identities. These are equations that are true for all replacements of the variable(s) for which both sides are defined. We now consider another class of trigonometric equations, called conditional equations, which may be true for some replacements of the variable but false for others. For example, cos x ⫽ sin x is a conditional equation, since it is true for some values, for example, x ⫽ /4, and false for others, such as x ⫽ 0. (Check both values.) This section considers two approaches for solving conditional trigonometric equations: an algebraic approach and a graphing utility approach. Solving trigonometric equations using an algebraic approach often requires the use of algebraic manipulation, identities, and ingenuity. In some cases algebraic methods lead to exact solutions, which are very useful in certain contexts. Graphing utility methods can be used to approximate solutions to a greater variety of trigonometric equations, but usually do not produce exact solutions. Each method has its strengths. Explore/Discuss 1 We are interested in solutions to the equation cos x ⫽ 0.5 The figure at the top of the next page shows a partial graph of the left and right sides of the equation. 486 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Explore/Discuss y y ⫽ cos x 1 1 y ⫽ 0.5 continued ⫺4 x 0 ⫺2 2 4 ⫺1 (A) How many solutions does the equation have on the interval [0, 2)? What are they? (B) How many solutions does the equation have on the interval (⫺⬁, ⬁)? Discuss a method of writing all solutions to the equation. Solving Trigonometric Equations Using an Algebraic Approach You might find the following suggestions for solving trigonometric equations using an algebraic approach useful: SUGGESTIONS FOR SOLVING TRIGONOMETRIC EQUATIONS ALGEBRAICALLY 1. Regard one particular trigonometric function as a variable, and solve for it. (A) Consider using algebraic manipulation such as factoring, combining or separating fractions, and so on. (B) Consider using identities. 2. After solving for a trigonometric function, solve for the variable. A number of examples should help make the algebraic approach clear. EXAMPLE Exact Solutions Using Factoring 1 Find all solutions exactly for 2 cos2 x ⫺ cos x ⫽ 0. Step 1. Solve for cos x. Solution 2 cos2 x ⫺ cos x ⫽ 0 FIGURE 1 cos x (2 cos x ⫺ 1) ⫽ 0 y cos x ⫽ 0 1 1/2 0 ⫺1 x 2 or Factor out cos x. ab ⫽ 0 only if a ⫽ 0 or b ⫽ 0 2 cos x ⫺ 1 ⫽ 0 cos x ⫽ 12 Step 2. Solve each equation over one period [0, 2). Sketch a graph of y ⫽ cos x, y ⫽ 0, and y ⫽ 12 in the same coordinate system to provide an aid to writing all solutions over one period (Fig. 1). 487 6-5 Trigonometric Equations cos x ⫽ 12 cos x ⫽ 0 x ⫽ /2, 3/2 x ⫽ /3, 5/3 Step 3. Write an expression for all solutions. Since the cosine function is peri- odic with period 2, all solutions are given by: 冦 /3 ⴙ 2k /2 ⴙ 2k xⴝ 3/2 ⴙ 2k 5/3 ⴙ 2k MATCHED PROBLEM k any integer Find all solutions exactly for 2 sin2 x ⫹ sin x ⫽ 0. 1 EXAMPLE Approximate Solutions Using Identities and Factoring 2 Find all real solutions for 3 cos2 x ⫹ 8 sin x ⫽ 7. Compute all inverse functions to four decimal places. Step 1. Solve for sin x and/or cos x. Move all nonzero terms to the left of the Solution equal sign and express the left side in terms of sin x: 3 cos2 x ⫹ 8 sin x ⫽ 7 3 cos2 x ⫹ 8 sin x ⫺ 7 ⫽ 0 cos2 x ⫽ 1 ⫺ sin2 x 3(1 ⫺ sin2 x) ⫹ 8 sin x ⫺ 7 ⫽ 0 3 sin2 x ⫺ 8 sin x ⫹ 4 ⫽ 0 (sin x ⫺ 2)(3 sin x ⫺ 2) ⫽ 0 sin x ⫺ 2 ⫽ 0 3u2 ⫺ 8u ⫹ 4 ⫽ (u ⫺ 2)(3u ⫺ 2) ab ⫽ 0 only if a ⫽ 0 or b ⫽ 0 or 3 sin x ⫺ 2 ⫽ 0 sin x ⫽ 2 y ⫽ sin x, y ⫽ 2, and y ⫽ 23 in the same coordinate system to provide an aid to writing all solutions over one period (Fig. 2). Solve the first equation: y 2 sin x ⫽ 2 1 2/3 2 ⫺1 2 3 Step 2. Solve each equation over one period [0, 2): Sketch a graph of FIGURE 2 0 sin x ⫽ x No solution, since ⫺1 ⱕ sin x ⱕ 1. Solve the second equation: sin x ⫽ 23 From the graph we see there are solutions in the first and second quadrants. x ⫽ sin⫺1 23 ⫽ 0.7297 First quadrant solution x ⫽ ⫺ 0.7297 ⫽ 2.4119 Second quadrant solution 488 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS sin 0.7297 ⫽ .6667; sin 2.4119 ⫽ 0.6666 (Checks may not be exact because of roundoff errors.) Step 3. Write an expression for all solutions. Since the sine function is periodic with period 2, all solutions are given by: Check xⴝ 0.7297 ⴙ 2k 冦 2.4119 ⴙ 2k k any integer Find all real solutions to 8 sin2 x ⫽ 5 ⫺ 10 cos x. Compute all inverse functions to four decimal places. MATCHED PROBLEM 2 EXAMPLE Approximate Solutions Using Substitution 3 Find in degree measure to three decimal places so that 5 sin (2 ⫺ 5) ⫽ ⫺3.045, 0° ⱕ 2 ⫺ 5 ⱕ 360°. Step 1. Make a substitution. Let u ⫽ 2 ⫺ 5 to obtain Solution 5 sin u ⫽ ⫺3.045, 0° ⱕ u ⱕ 360° Step 2. Solve for sin u. sin u ⫽ ⫺3.045 ⫽ ⫺0.609 5 Step 3. Solve for u over 0° ⱕ u ⱕ 360°. Sketch a graph of y ⫽ sin u and FIGURE 3 y ⫽ ⫺0.609 in the same coordinate system to provide an aid to writing all solutions over 0° ⱕ u ⱕ 360° (Fig. 3). Solutions are in the third and fourth quadrants. If the reference angle is ␣, then u ⫽ 180° ⫹ ␣ or u ⫽ 360° ⫺ ␣. y 1 180⬚ 0 360⬚ x ␣ ⫽ sin⫺1 0.609 ⫽ 37.571° Reference angle u ⫽ 180° ⫹ 37.571° ⫺0.609 ⫽ 217.517° ⫺1 Third quadrant solution u ⫽ 360° ⫺ 37.571° ⫽ 322.429° Check Fourth quadrant solution sin 217.517° ⫽ ⫺0.609; sin 322.429° ⫽ ⫺0.610 Step 4. Now solve for : u ⫽ 217.517° u ⫽ 322.429° 2 ⫺ 5 ⫽ 217.517° 2 ⫺ 5 ⫽ 322.429° ⴝ 111.259° ⴝ 163.715° A final check in the original equation is left to the reader. 6-5 Trigonometric Equations MATCHED PROBLEM 3 EXAMPLE 4 Solution 489 Find in degree measure to three decimal places so that 8 tan (6 ⫹ 15) ⫽ ⫺64.328, ⫺90° ⬍ 6 ⫹ 15 ⬍ 90°. Exact Solutions Using Identities and Factoring Find exact solutions for sin2 x ⫽ 1 2 sin 2x, 0 ⱕ x ⬍ 2. The following solution includes only the key steps. Sketch graphs as appropriate on scratch paper. sin2 x ⫽ 12 sin 2x Use double-angle identity ⫽ (2 sin x cos x) 1 2 sin2 x ⫺ sin x cos x ⫽ 0 a2 ⫺ ab ⫽ a(a ⫺ b) sin x (sin x ⫺ cos x) ⫽ 0 sin x ⫽ 0 a(a ⫺ b) ⫽ 0 only if a ⫽ 0 or a ⫺ b ⫽ 0 or sin x ⫺ cos x ⫽ 0 x ⫽ 0, sin x ⫽ cos x sin x ⫽1 cos x tan x ⫽ 1 x ⫽ /4, 5/4 Combining the solutions from both equations, we have the complete set of solutions: x ⴝ 0, /4, , 5/4 MATCHED PROBLEM Find exact solutions for sin 2x ⫽ sin x, 0 ⱕ x ⬍ 2. 4 EXAMPLE Approximate Solutions Using Identities and the Quadratic Formula 5 Solve cos 2x ⫽ 4 cos x ⫺ 2 for all real x. Compute inverse functions to four decimal places. Solution Step 1. Solve for cos x. cos 2x ⫽ 4 cos x ⫺ 2 Use double-angle identity. 2 cos2 x ⫺ 1 ⫽ 4 cos x ⫺ 2 2 cos2 x ⫺ 4 cos x ⫹ 1 ⫽ 0 Quadratic in cos x. Left side does not factor using integer coefficients. Solve using quadratic formula. 490 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS cos x ⫽ 4 ⫾ 兹16 ⫺ 4(2)(1) 2(2) ⫽ 1.707107 or 0.292893 Step 2. Solve each equation over one period [0, 2): Sketch a graph of y ⫽ cos x, y ⫽ 1.707107, and y ⫽ 0.292893 in the same coordinate system to provide an aid to writing all solutions over one period (Fig. 4). FIGURE 4 y 2 1.707107 1 0.292893 0 x 2 ⫺1 Solve the first equation: cos x ⫽ 1.707107 No solution, since ⫺1 ⱕ cos x ⱕ 1 Solve the second equation: cos x ⫽ 0.292893 Figure 4 indicates a first quadrant solution and a fourth quadrant solution. If the reference angle is ␣, then x ⫽ ␣ or x ⫽ 2 ⫺ ␣. ␣ ⫽ cos⫺1 0.292893 ⫽ 1.2735 2 ⫺ ␣ ⫽ 2 ⫺ 1.2735 ⫽ 5.0096 Check cos 1.2735 ⫽ 0.292936; cos 5.0096 ⫽ 0.292854 Step 3. Write an expression for all solutions. Since the cosine function is peri- odic with period 2, all solutions are given by: xⴝ MATCHED PROBLEM 5 1.2735 ⴙ 2k 冦 5.0096 ⴙ 2k k any integer Solve cos 2x ⫽ 2(sin x ⫺ 1) for all real x. Compute inverse functions to four decimal places. Solving Trigonometric Equations Using a Graphing Utility All the trigonometric equations that were solved earlier with algebraic methods can also be solved, though usually not exactly, with graphing utility methods. In addition, there are many trigonometric equations that can be solved (to any decimal accuracy desired) using graphing utility methods, but cannot be solved in a finite sequence of steps using algebraic methods. Examples 6–8 are such examples. 6-5 Trigonometric Equations EXAMPLE 491 Solution Using a Graphing Utility 6 Find all real solutions to four decimal places for 2 cos 2x ⫽ 1.35x ⫺ 2. Solution This relatively simple trigonometric equation cannot be solved using a finite number of algebraic steps (try it!). However, it can be solved rather easily to the accuracy desired using a graphing utility. Graph y1 ⫽ 2 cos 2x and y2 ⫽ 1.35x ⫺ 2 in the same viewing window, and find any points of intersection using a built-in routine. The first point of intersection is shown in Figure 5. It appears there may be more than one point of intersection, but zooming in on the portion of the graph in question shows that the two graphs do not intersect in that region (Fig. 6). The only solution is x ⫽ 0.9639 Left side: 2 cos 2(0.9639) ⫽ ⫺0.6989 Right side: 1.35(0.9639) ⫺ 2 ⫽ ⫺0.6987 Check 3 4 ⫺2 /2 2 3/2 ⫺4 1 FIGURE 5 MATCHED PROBLEM FIGURE 6 Find all real solutions to four decimal places for sin x/2 ⫽ 0.2x ⫺ 0.5. 6 EXAMPLE 7 Geometric Application A 10-cm arc on a circle has an 8-cm chord. What is the radius of the circle to four decimal places? What is the radian measure of the central angle, to four decimal places, subtended by the arc? Solution Sketch a figure with auxiliary lines (Fig. 7). From the figure, in radians is FIGURE 7 5 5 4 4 R ⫽ R 5 R and sin ⫽ 4 R Thus, sin 4 5 ⫽ R R and our problem is to solve this trigonometric equation for R. Algebraic methods will not isolate R, so turn to the use of a graphing utility. Start by graphing y1 ⫽ sin 5/x and y2 ⫽ 4/x in the same viewing window for 1 ⱕ x ⱕ 10 and 492 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS ⫺2 ⱕ y ⱕ 2 (Fig. 8). It appears that the graphs intersect for x between 4 and 5. To get a clearer look at the intersection point we change the window dimensions to 4 ⱕ x ⱕ 5 and 0.5 ⱕ y ⱕ 1.5, and use a built-in routine to find the point of intersection (Fig. 9). 1.5 2 1 10 4 5 ⫺2 0.5 FIGURE 8 FIGURE 9 From Figure 9, we see that R ⴝ 4.4205 cm sin 5/R ⫽ sin (5/4.4205) ⫽ 0.9049; 4/R ⫽ 4/4.4205 ⫽ 0.9049 Check Having R, we can compute the radian measure of the central angle subtended by the 10-cm arc: Central Angle ⴝ MATCHED PROBLEM 7 An 8.2456-in. arc on a circle has a 6.0344-in. chord. What is the radius of the circle to four decimal places? What is the measure of the central angle, to four decimal places, subtended by the arc? EXAMPLE Solution Using a Graphing Utility 8 Solution 10 10 ⴝ ⴝ 2.2622 rad R 4.4205 Find all real solutions, to four decimal places, for tan (x/2) ⫽ 1/x, ⫺ ⬍ x ⱕ 3. Graph y1 ⫽ tan (x/2) and y2 ⫽ 1/x in the same viewing window for ⫺ ⬍ x ⬍ 3 (Fig. 10). Solutions are at points of intersection. FIGURE 10 2 ⫺ 3 ⫺2 Using a built-in routine, the three solutions are found to be x ⫽ ⫺1.3065, 1.3065, 6.5846 Checking these solutions is left to the reader. 493 6-5 Trigonometric Equations MATCHED PROBLEM 8 Find all real solutions, to four decimal places, for 0.25 tan (x/2) ⫽ ln x, 0 ⬍ x ⬍ 4. Solving trigonometric inequalities using a graphing utility is as easy as solving trigonometric equations using a graphing utility. Example 9 illustrates the process. EXAMPLE Solving a Trigonometric Inequality 9 Solve sin x ⫺ cos x ⬍ 0.25x ⫺ 0.5, using two-decimal-place accuracy. Graph y1 ⫽ sin x ⫺ cos x and y2 ⫽ 0.25x ⫺ 0.5 in the same viewing window (Fig. 11). Solution FIGURE 11 2 ⫺2 3 ⫺2 Finding the three points of intersection by a built-in routine, we see that the graph of y1 is below the graph of y2 on the following two intervals: (⫺1.65, 0.52) and (3.63, ⬁). Thus, the solution set to the inequality is (⫺1.65, 0.52) 艛 (3.63, ⬁). MATCHED PROBLEM Solve cos x ⫺ sin x ⬎ 0.4 ⫺ 0.3x, using two-decimal-place accuracy. 9 Explore/Discuss How many solutions does the following equation have? sin (1/x) ⫽ 0 2 (1) Graph y1 ⫽ sin (1/x) and y2 ⫽ 0 for each of the indicated intervals in parts A–G. From each graph estimate the number of solutions that equation (1) appears to have. What final conjecture would you be willing to make regarding the number of solutions to equation (1)? Explain. (A) [⫺20, 20]; Can 0 be a solution? Explain. (B) [⫺2, 2] (C) [⫺1, 1] (D) [⫺0.1, 0.1] (E) [⫺0.01, 0.01] (F) [⫺0.001, 0.001] (G) [⫺0.0001, 0.0001] Answers to Matched Problems 冦 0 ⫹ 2k ⫹ 2k 1. x ⫽ 7/6 ⫹ 2k 11/6 ⫹ 2k k any integer 4. x ⫽ 0, /3, , 5/3 5. x ⫽ 2. x ⫽ 0.9665 ⫹ 2k 冦 2.1751 ⫹ 2k 7. R ⫽ 3.1103 in; central angle ⫽ 2.6511 rad 1.8235 ⫹ 2k 冦 4.4597 ⫹ 2k k any integer k any integer 6. x ⫽ 5.1609 8. x ⫽ 1.1828, 2.6369, 9.2004 3. ⫺16.318° 9. (⫺1.67, 0.64) 艛 (3.46, ⬁) 494 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS EXERCISE 6-5 26. cos x ⫽ cot x, 0 ⱕ x ⬍ 2 27. sec (x/2) ⫹ 2 ⫽ 0, 0 ⱕ x ⬍ 2 A In Problems 1–12, find exact solutions over the indicated intervals (x a real number, in degrees). 28. tan (x/2) ⫺ 1 ⫽ 0, 0 ⱕ x ⬍ 2 29. 2 cos2 ⫹ 3 sin ⫽ 0, 0° ⱕ ⬍ 360° 30. sin2 ⫹ 2 cos ⫽ ⫺2, 0° ⱕ ⬍ 360° 1. 2 sin x ⫹ 1 ⫽ 0, 0 ⱕ x ⬍ 2 31. cos 2 ⫹ cos ⫽ 0, 0° ⱕ ⬍ 360° 2. 2 cos x ⫹ 1 ⫽ 0, 0 ⱕ x ⬍ 2 32. cos 2 ⫹ sin2 ⫽ 0, 0° ⱕ ⬍ 360° 3. 2 sin x ⫹ 1 ⫽ 0, all real x 33. 2 sin2 (x/2) ⫺ 3 sin (x/2) ⫹ 1 ⫽ 0, 0 ⱕ x ⱕ 2 4. 2 cos x ⫹ 1 ⫽ 0, all real x 34. 4 cos2 2x ⫺ 4 cos 2x ⫹ 1 ⫽ 0, 0 ⱕ x ⱕ 2 5. tan x ⫹ 兹3 ⫽ 0, 0 ⱕ x ⬍ 6. 兹3 tan x ⫹ 1 ⫽ 0, 0 ⱕ x ⬍ 7. tan x ⫹ 兹3 ⫽ 0, all real x 8. 兹3 tan x ⫹ 1 ⫽ 0, all real x 9. 2 cos ⫺ 兹3 ⫽ 0, 0° ⱕ ⬍ 360° 10. 兹2 sin ⫺ 1 ⫽ 0, 0° ⱕ ⬍ 360° 11. 2 cos ⫺ 兹3 ⫽ 0, all 12. 兹2 sin ⫺ 1 ⫽ 0, all Solve Problems 13–18 to four decimal places ( in degrees, x real). 13. 7 cos x ⫺ 3 ⫽ 0, 0 ⱕ x ⬍ 2 14. 5 cos x ⫺ 2 ⫽ 0, 0 ⱕ x ⬍ 2 15. 2 tan ⫺ 7 ⫽ 0, 0° ⱕ ⬍ 180° 16. 4 tan ⫹ 15 ⫽ 0, 0° ⱕ ⬍ 180° 17. 1.3224 sin x ⫹ 0.4732 ⫽ 0, all real x 18. 5.0118 sin x ⫺ 3.1105 ⫽ 0, all real x Solve problems 19–22 to four decimal places using a graphing utility. 19. 1 ⫺ x ⫽ 2 sin x, all real x 20. 2x ⫺ cos x ⫽ 0, all real x 21. tan (x/2) ⫽ 8 ⫺ x, 0 ⱕ x ⬍ 22. tan 2x ⫽ 1 ⫹ 3x, 0 ⱕ x ⬍ /4 Solve Problems 35–40 (x real and in degrees). Compute inverse functions to four significant digits. 35. 6 sin2 ⫹ 5 sin ⫽ 6, 0° ⱕ ⱕ 90° 36. 4 cos2 ⫽ 7 cos ⫹ 2, 0° ⱕ ⱕ 180° 37. 3 cos2 x ⫺ 8 cos x ⫽ 3, 0 ⱕ x ⱕ 38. 8 sin2 x ⫹ 10 sin x ⫽ 3, 0 ⱕ x ⱕ /2 39. 2 sin x ⫽ cos 2x, 0 ⱕ x ⬍ 2 40. cos 2x ⫹ 10 cos x ⫽ 5, 0 ⱕ x ⬍ 2 Solve Problems 41 and 42 for all real number solutions. Compute inverse functions to four significant digits. 41. 2 sin2 x ⫽ 1 ⫺ 2 sin x 42. cos2 x ⫽ 3 ⫺ 5 cos x Solve Problems 43–52 to four decimal places using a graphing utility. 43. 2 sin x ⫽ cos 2x, 0 ⱕ x ⬍ 2 44. cos 2x ⫹ 10 cos x ⫽ 5, 0 ⱕ x ⬍ 2 45. 2 sin2 x ⫽ 1 ⫺ 2 sin x, all real x 46. cos2 x ⫽ 3 ⫺ 5 cos x, all real x 47. cos 2x ⬎ x2 ⫺ 2, all real x 48. 2 sin (x ⫺ 2) ⬍ 3 ⫺ x2, all real x 49. cos (2x ⫹ 1) ⱕ 0.5x ⫺ 2, all real x 50. sin (3 ⫺ 2x) ⱖ 1 ⫺ 0.4x, all real x B 51. esin x ⫽ 2x ⫺ 1, all real x In Problems 23–34, find exact solutions for x real and in degrees. 52. e⫺sin x ⫽ 3 ⫺ x, all real x 23. 2 sin2 ⫹ sin 2 ⫽ 0, all 24. cos2 ⫽ 12 sin 2, all 25. tan x ⫽ ⫺2 sin x, 0 ⱕ x ⬍ 2 53. Explain the difference between evaluating tan⫺1 (⫺5.377) and solving the equation tan x ⫽ ⫺5.377. 54. Explain the difference between evaluating cos⫺1 (⫺0.7334) and solving the equation cos x ⫽ ⫺0.7334. 495 6-5 Trigonometric Equations C Find exact solutions to Problems 55–58. [Hint: Square both sides at an appropriate point, solve, then eliminate extraneous solutions at the end.] 55. cos x ⫺ sin x ⫽ 1, 0 ⱕ x ⬍ 2 56. sin x ⫹ cos x ⫽ 1, 0 ⱕ x ⬍ 2 57. tan x ⫺ sec x ⫽ 1, 0 ⱕ x ⬍ 2 64. Electric Current. Refer to Problem 63. Find the smallest positive t (to four significant digits) such that I ⫽ 25 amperes. 65. Optics. A polarizing filter for a camera contains two parallel plates of polarizing glass, one fixed and the other able to rotate. If is the angle of rotation from the position of maximum light transmission, then the intensity of light leaving the filter is cos2 times the intensity I of light entering the filter (see the figure). 58. sec x ⫹ tan x ⫽ 1, 0 ⱕ x ⬍ 2 Polarizing filter (schematic) Solve Problems 59–60 to four significant digits using a graphing utility. 59. sin (1/x) ⫽ 1.5 ⫺ 5x, 0.04 ⱕ x ⱕ 0.2 Light 60. 2 cos (1/x) ⫽ 950x ⫺ 4, 0.006 ⬍ x ⬍ 0.007 I 61. We are interested in the zeros of the function f(x) ⫽ sin (1/x) for x ⬎ 0. (A) Explore the graph of f over different intervals [0.1, b] for various values of b, b ⬎ 0.1. Does the function f have a largest zero? If so, what is it (to four decimal places)? Explain what happens to the graph of f as x increases without bound. Does the graph have an asymptote? If so, what is its equation? (B) Explore the graph of f over different intervals (0, b] for various values of b, 0 ⬍ b ⱕ 0.1. How many zeros exist between 0 and b, for any b ⬎ 0, however small? Explain why this happens. Does f have a smallest positive zero? Explain. 62. We are interested in the zeros of the function g(x) ⫽ cos (1/x) for x ⬎ 0. (A) Explore the graph of g over different intervals [0.1, b] for various values of b, b ⬎ 0.1. Does the function g have a largest zero? If so, what is it (to four decimal places)? Explain what happens to the graph of g as x increases without bound. Does the graph have an asymptote? If so, what is its equation? (B) Explore the graph of g over different intervals (0, b] for various values of b, 0 ⬍ b ⱕ 0.1. How many zeros exist between 0 and b, for any b ⬎ 0, however small? Explain why this happens. Does g have a smallest positive zero? Explain. I cos 2 Find the smallest positive (in decimal degrees to two decimal places) so that the intensity of light leaving the filter is 40% of that entering. 66. Optics. Refer to Problem 65. Find the smallest positive so that the light leaving the filter is 70% of that entering. 67. Astronomy. The planet Mercury travels around the sun in an elliptical orbit given approximately by r⫽ 3.44 ⫻ 107 1 ⫺ 0.206 cos (see the figure). Find the smallest positive (in decimal degrees to three significant digits) such that Mercury is 3.09 ⫻ 107 miles from the sun. Mercury r Sun APPLICATIONS Orbit 63. Electric Current. An alternating current generator produces a current given by the equation I ⫽ 30 sin 120t where t is time in seconds and I is current in amperes. Find the smallest positive t (to four significant digits) such that I ⫽ ⫺10 amperes. 68. Astronomy. Refer to Problem 67. Find the smallest positive (in decimal degrees to three significant digits) such that Mercury is 3.78 ⫻ 107 miles from the sun. 496 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS 69. Geometry. The area of the segment of a circle in the figure is given by A ⫽ 12 R2 ( ⫺ sin ) R a where is in radian measure. Use a graphing utility to find the radian measure, to three decimal places, of angle , if the radius is 8 inches and the area of the segment is 48 square inches. b L Cornea R 72. Eye Surgery. Refer to Problem 71. (A) If in the figure a ⫽ 5.4 millimeters and b ⫽ 2.4 millimeters, find L correct to four decimal places. (B) Increasing the chord length without changing the arc length L has the effect of pulling the cornea inward and giving it a flatter, yet still circular, shape. With the aid of a graphing utility in part of the solution, approximate b to four decimal places if a is increased to 5.5 millimeters and L remains the same as it was in part A. R 70. Geometry. Repeat Problem 69, if the radius is 10 centimeters and the area of the segment is 40 square centimeters. 71. Eye Surgery. A surgical technique for correcting an astigmatism involves removing small pieces of tissue in order to change the curvature of the cornea.* In the cross section of a cornea shown in the figure, the circular arc, with radius R and central angle 2, represents a cross section of the surface of the cornea. (A) If a ⫽ 5.5 millimeters and b ⫽ 2.5 millimeters, find L correct to four decimal places. (B) Reducing the chord length 2a without changing the length L of the arc has the effect of pushing the cornea outward and giving it a rounder, yet still a circular, shape. With the aid of a graphing utility in part of the solution, approximate b to four decimal places if a is reduced to 5.4 millimeters and L remains the same as it was in part A. *Based on the article “The Surgical Correction of Astigmatism” by Sheldon Rothman and Helen Strassberg in the UMAP Journal, Vol. v, no. 2, 1984. Analytic Geometry. Find simultaneous solutions for each system of equations in Problems 73 and 74 (0° ⱕ ⱕ 360°). These are polar equations, which will be discussed in the next chapter. ★ 73. r ⫽ 2 sin r ⫽ sin 2 ★ 74. r ⫽ 2 sin r ⫽ 2(1 ⫺ sin ) Problems 75 and 76 are related to rotation of axes in analytic geometry. ★★ 75. Analytic Geometry. Given the equation 2xy ⫽ 1, replace x and y with x ⫽ u cos ⫺ v sin y ⫽ u sin ⫹ v cos and simplify the left side of the resulting equation. Find the smallest positive in degree measure so that the coefficient of the uv term is 0. ★★ 76. Analytic Geometry. Repeat Problem 75 for xy ⫽ ⫺2.
