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All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)
Double Angle Formulas Find sin 2x, cos 2x, and tan 2x from the given information. 4. tanx=43, x in Quadrant II5E6E7E8E9E10E11E12E13ELowering Powers in a Trigonometric Expression Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine, as in Example 4. 14. cos4 x sin2 x15ELowering Powers in a Trigonometric Expression Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine, as in Example 4. 16. cos6 x17E18EHalf Angle Formulas Use an appropriate Half-Angle Formula to find the exact value of the expression. 19. tan 22.520E21E22E23E24E25E26E27E28EDouble- and Half-Angle Formulas Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. 29. (a) 2 sin 18 cos 18 (b) 2 sin 3 cos 3Double- and Half-Angle Formulas Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. 30. (a) 2tan71tan27 (b) 2tan71tan27Double- and Half-Angle Formulas Simplify the expression by using a Double-Angle Formula or a Half-Angle Formula. 31. (a) cos2 34 sin2 34 (b) cos2 5 sin2 532E33E34EProving a Double-Angle Formula Use the Addition Formula for Sine to prove the Double-Angle Formula for Sine.36EUsing a Half-Angle Formula Find sinx2,cosx2, and tanx2 from the given information. 37. sinx=35, 0 x 9038E39E40E41E42E43E44E45E46E47E48E49E50EEvaluating an Expression Involving Trigonometric Functions Evaluate each expression under the given conditions. 51. cos 2; sin=35, in Quadrant III52EEvaluating an Expression Involving Trigonometric Functions Evaluate each expression under the given conditions. 53. sin 2; sin=17, in Quadrant IIEvaluating an Expression Involving Trigonometric Functions Evaluate each expression under the given conditions. 54. tan 2; cos=35, in Quadrant I55E56E57E58E59E60E61E62E63ESum-to-Product Formulas Write the sum as a product. 64. cos 9x + cos 2x65E66E67E68EValue of a Product or Sum Find the value of the product or sum. 69. cos 37.5 sin 7.570EValue of a Product or Sum Find the value of the product or sum. 71. cos 255 cos 19572E73EProving Identities Prove the identity. 74. sin 8x = 2 sin 4x cos 4x75E76E77E78E79E80E81E82E83E84E85EProving Identities Prove the identity. 86. 4(sin6 x + cos6 x) = 4 3 sin2 2x87E88E89E90E91E92E93E94E95E96ESum-to-Product Formulas Use a Sum-to-Product Formula to show the following. 97. sin 130 sin 110 = sin 10Sum-to-Product Formulas Use a Sum-to-Product Formula to show the following. 98. cos 100 cos 200 = sin 5099ESum-to-Product Formulas Use a Sum-to-Product Formula to show the following. 100. cos 87 + cos 33 = sin 63101E102E103E104E105E106E107E108E109ELength of a Bisector In triangle ABC (see the figure) the line segment s bisects angle C. Show that the length of s is given by s=2abcosxa+b [Hint: Use the Law of Sines.]111ELargest Area A rectangle is to be inscribed in a semicircle of radius 5 cm as shown in the following figure. (a) Show that the area of the rectangle is modeled by the function A()=25sin2 (b) Find the largest possible area for such an inscribed rectangle. [Hint: Use the fact that sin u achieves its maximum value at u = /2.] (c) Find the dimensions of the inscribed rectangle with the largest possible area.Sawing a Wooden Beam A rectangular beam is to be cut from a cylindrical log of diameter 20 in. (a) Show that the cross-sectional area of the beam is modeled by the function A()=200sin2where is as shown in the figure. (b) Show that the maximum cross-sectional area of such a beam is 200 in2. [Hint: Use the fact that sin u achieves its maximum value at u = /2.]114E115ETouch-Tone Telephones When a key is pressed on a touchtone telephone, the keypad generates two pure tones, which combine to produce a sound that uniquely identifies the key. The figure shows the low frequency f1 and the high frequency f2 associated with each key. Pressing a key produces the sound wave y = sin(2f1t) + sin(2f2t). (a) Find the function that models the sound produced when the 4 key is pressed. (b) Use a Sum-to-Product Formula to express the sound generated by the 4 key as a product of a sine and a cosine function. (c) Graph the sound wave generated by the 4 key from t = 0 to t = 0.006 s.117EBecause the trigonometric functions are periodic, if a basic trigonometric equation has one solution, it has _____ (several/infinitely many) solutions.The basic equation sin x = 2 has _____ (no/one/infinitely many) solutions, whereas the basic equation sin x = 0.3 has _____ (no/one/infinitely many) solutions.We can find some of the solutions of sin x = 0.3 graphically by graphing y = sin x and y = _____. Use the graph below to estimate some of the solutions.4E5E6E7E8E9E10E11E12E13E14E15E16ESolving Basic Trigonometric Equations Solve the given equation, and list six specific solutions. 17. cos=32Solving Basic Trigonometric Equations Solve the given equation, and list six specific solutions. 18. cos=1219E20E21ESolving Basic Trigonometric Equations Solve the given equation, and list six specific solutions. 22. tan = 2.523ESolving Basic Trigonometric Equations Solve the given equation, and list six specific solutions. 24. sin = 0.925E26E27E28E29E30E31ESolving Trigonometric Equations Find all solutions of the given equation. 32. cot + 1 = 033E34E35E36E37E38E39ESolving Trigonometric Equations by Factoring Solve the given equation. 40. (tan 2)(16 sin2 1) = 041E42E43ESolving Trigonometric Equations by Factoring Solve the given equation. 44. tan4 13 tan2 + 36 = 0Solving Trigonometric Equations by Factoring Solve the given equation. 45. 2 cos2 7 cos + 3 = 046E47E48E49E50E51E52E53E54ESolving Trigonometric Equations by Factoring Solve the given equation. 55. 3 tan sin 2 tan = 056ERefraction of Light It has been observed since ancient times that light refracts, or bends, as it travels from one medium to another (from air to water, for example). If v1, is the speed of light in one medium and v2 is its speed in another medium, then according to Snells Law, sin1sin2=v1v2 where 1 is the angle of incidence and 2 is the angle of refraction (see the figure). The number v1/v2 is called the index of refraction. The index of refraction for several substances is given in the table. If a ray of light passes through the surface of a lake at an angle of incidence of 70, what is the angle of refraction? Substance Refraction from air to substance Water 1.33 Alcohol 1.36 Glass 1.52 Diamond 2.41Total Internal Reflection When light passes from a more-dense to a less-dense mediumfrom glass to air, for examplethe angle of refraction predicted by Snells Law (see Exercise 57) can be 90 or larger. In this case the light beam is actually reflected back into the denser medium. This phenomenon, called total internal reflection, is the principle behind fiber optics. Set 2 = 90 in Snells Law, and solve for 1 to determine the critical angle of incidence at which total internal reflection begins to occur when light passes from glass to air. (Note that the index of refraction from glass to air is the reciprocal of the index from air to glass.)Phases of the Moon As the moon revolves around the earth, the side that faces the earth is usually just partially illuminated by the sun. The phases of the moon describe how much of the surface appears to be in sunlight. An astronomical measure of phase is given by the fraction F of the lunar disc that is lit. When the angle between the sun, earth, and moon is (0 360), then F=12(1cos) Determine the angles that correspond to the following phases: (a) F = 0 (new moon) (b) F = 0.25 (a crescent moon) (c) F = 0.5 (first or last quarter) (d) F = 1 (full moon)60EWe can use identities to help us solve trigonometric equations. 1. Using a Pythagorean identity we see that the equation sin x + sin2x + cos2x = 1 is equivalent to the basic equation __________ whose solutions are x = _____.We can use identities to help us solve trigonometric equations. 2. Using a Double-Angle Formula we see that the equation sin x + sin 2x = 0 is equivalent to the equation _____. Factoring, we see that solving this equation is equivalent to solving the two basic equations _____ and _____.3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22ESolving Trigonometric Equations Involving a Multiple of an Angle An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval [0, 2). 23. cos21=024E25E26ESolving Trigonometric Equations Involving a Multiple of an Angle An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval [0, 2). 27. sin 2 = 3 cos 2Solving Trigonometric Equations Involving a Multiple of an Angle An equation is given. (a) Find all solutions of the equation. (b) Find the solutions in the interval [0, 2). 28. csc 3 = 5 sin 329E30E31ESolving Trigonometric Equations Solve the equations by factoring. 32. 4 sin cos + 2 sin 2 cos 1 = 033ESolving Trigonometric Equations Solve the equations by factoring. 34. sec tan cos cot = sin35E36E37EFinding Intersection Points Graphically (a) Graph f and g in the given viewing rectangle and find the intersection points graphically, rounded to two decimal places. (b) Find the intersection points of f and g algebraically. Give exact answers. 38. f(x) = sin x 1, g(x) = cos x; [2, 2] by [2.5, 1.5]39EUsing Addition or Subtraction Formulas Use an Addition or Subtraction Formula to simplify the equation. Then find all solutions in the interval [0, 2). 40. coscos2+sinsin2=1241EUsing Addition or Subtraction Formulas Use an Addition or Subtraction Formula to simplify the equation. Then find all solutions in the interval [0, 2). 42. sin 3 cos cos 3 sin = 043E44E45E46E47E48E49E50E51E52E53EUsing Sum-to-Product Formulas Solve the equation by first using a Sum-to-Product Formula. 54. cos 5 cos 7 = 055E56E57E58E59ESolving Trigonometric Equations Graphically Use a graphing device to find the solutions of the equation, rounded to two decimal places. 60. sin x = x361E62EEquations Involving Inverse Trigonometric Functions Solve the given equation for x. 63. tan1x+tan12x=4 [Hint: Let u = tan1 x and v = tan1 2x. Solve the equation u+v=4 by taking the tangent of each side.]Equations Involving Inverse Trigonometric Functions Solve the given equation for x. 64. 2 sin1 x + cos1 x = [Hint: Take the cosine of each side.]Range of a Projectile If a projectile is fired with velocity v0 at an angle , then its range, the horizontal distance it travels (in ft), is modeled by the function R()=v02sin232 (See page 627.) If v0 = 2200 ft/s, what angle (in degrees) should be chosen for the projectile to hit a target on the ground 5000 ft away?Damped Vibrations The displacement of a spring vibrating in damped harmonic motion is given by y=4e3tsin2t Find the times when the spring is at its equilibrium position (y = 0).Hours of Daylight In Philadelphia the number of hours of daylight on day t (where t is the number of days after January 1) is modeled by the function L(t)=12+2.83sin(2365(t80)) (a) Which days of the year have about 10 h of daylight? (b) How many days of the year have more than 10 h of daylight?Belts and Pulleys A thin belt of length L surrounds two pulleys of radii R and r, as shown in the figure to the right. (a) Show that the angle (in rad) where the belt crosses itself satisfies the equation +2cot2=LR+r [Hint: Express L in terms of R, r, and by adding up the lengths of the curved and straight parts of the belt.] (b) Suppose that R = 2.42 ft, r = 1.21 ft, and L = 27.78 ft. Find by solving the equation in part (a) graphically. Express your answer both in radians and in degrees.69EWhat is an identity? What is a trigonometric identity?2RCC3RCC4RCC5RCC6RCC7RCC8RCC9RCC10RCC11RCC12RCC1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41RE42RE43RE44RE45RE