 # PC Lesson 5.1 (23-47 odds, 48-54)

```5-1 Trigonometric Identities
Simplify each expression.
23. csc x – cos x cot x
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27. SOLUTION: 29. SOLUTION: 31. cot x – cos3 x csc x
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5-1 Trigonometric Identities
31. cot x – cos3 x csc x
SOLUTION: Simplify each expression.
33. +
SOLUTION: 35. +
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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
.
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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
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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
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45. SOLUTION: 47. SOLUTION: Determine whether each parent trigonometric function shown is odd or even. Explain your reasoning.
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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
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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
54. sec x csc x – cos x csc x
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53. csc x tan2 x – sec2 x csc x
SOLUTION: 54. sec x csc x – cos x csc x   