5-1 Trigonometric Identities Simplify each expression. 23. csc x – cos x cot x SOLUTION: 25. SOLUTION: 27. SOLUTION: eSolutions Manual - Powered by Cognero Page 1 5-1 Trigonometric Identities 27. SOLUTION: 29. SOLUTION: 31. cot x – cos3 x csc x SOLUTION: eSolutions Manual - Powered by Cognero Page 2 5-1 Trigonometric Identities 31. cot x – cos3 x csc x SOLUTION: Simplify each expression. 33. + SOLUTION: 35. + SOLUTION: eSolutions Manual - Powered by Cognero Page 3 5-1 Trigonometric Identities 35. + SOLUTION: 37. SUNGLASSES Many sunglasses are made with polarized lenses, which reduce the intensity of light. The intensity of light emerging from a system of two polarizing lenses I can be calculated by I = I0 – , where I0 is the intensity of light entering the system of lenses and θ is the angle of the axis of the second lens in relation to that of the first lens. a. Simplify the formula for the intensity of light emerging from a system of two polarized lenses. b. If a pair of sunglasses contains a system of two polarizing lenses with axes at 30º to one another, what proportion of the intensity of light entering the sunglasses emerges? SOLUTION: a. b. Evaluate for eSolutions Manual - Powered by Cognero . Page 4 5-1 Trigonometric Identities 37. SUNGLASSES Many sunglasses are made with polarized lenses, which reduce the intensity of light. The intensity of light emerging from a system of two polarizing lenses I can be calculated by I = I0 – , where I0 is the intensity of light entering the system of lenses and θ is the angle of the axis of the second lens in relation to that of the first lens. a. Simplify the formula for the intensity of light emerging from a system of two polarized lenses. b. If a pair of sunglasses contains a system of two polarizing lenses with axes at 30º to one another, what proportion of the intensity of light entering the sunglasses emerges? SOLUTION: a. b. Evaluate Therefore, for . of the intensity of light entering the sunglasses emerges. Rewrite as an expression that does not involve a fraction. 39. SOLUTION: eSolutions Manual - Powered by Cognero Page 5 5-1 Therefore, Trigonometric Identities of the intensity of light entering the sunglasses emerges. Rewrite as an expression that does not involve a fraction. 39. SOLUTION: 41. SOLUTION: eSolutions Manual - Powered by Cognero 43. Page 6 5-1 Trigonometric Identities 41. SOLUTION: 43. SOLUTION: 45. SOLUTION: eSolutions Manual - Powered by Cognero Page 7 5-1 Trigonometric Identities 45. SOLUTION: 47. SOLUTION: Determine whether each parent trigonometric function shown is odd or even. Explain your reasoning. eSolutions Manual - Powered by Cognero Page 8 5-1 Trigonometric Identities Determine whether each parent trigonometric function shown is odd or even. Explain your reasoning. 48. SOLUTION: Odd; the graph shown is f (x) = cot x. f (–x) = cot (–x) = –cot x = –f (x). Since f (–x) = –f (x), f (x) = cot x is an odd function. 49. SOLUTION: Even; the graph shown is f (x) = sec x. f (–x) = sec (–x) = sec x = f (x). Since f (–x) = f (x), f (x) = sec x is an even function. 50. SOCCER When a soccer ball is kicked from the ground, its height y and horizontal displacement x are related by the equation y = + , where v0 is the initial velocity of the ball, kicked, and g is the acceleration due to gravity. Rewrite this equation so that tan that appears in the equation. is the angle at which it was is the only trigonometric function SOLUTION: Write each expression in terms of a single trigonometric function. 51. tan x – csc x sec x SOLUTION: eSolutions Manual - Powered by Cognero Page 9 5-1 Trigonometric Identities Write each expression in terms of a single trigonometric function. 51. tan x – csc x sec x SOLUTION: 52. cos x + tan x sin x SOLUTION: 53. csc x tan2 x – sec2 x csc x SOLUTION: eSolutions Manual - Powered by Cognero 54. sec x csc x – cos x csc x Page 10 5-1 Trigonometric Identities 53. csc x tan2 x – sec2 x csc x SOLUTION: 54. sec x csc x – cos x csc x SOLUTION: eSolutions Manual - Powered by Cognero Page 11

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