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If cos x = 2 3 , and x is in quadrant I.

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If cos x = 1 2 , and x is in quadrant III.

a) 3 2 b) 1 2 c) 3

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If tan x = 8 , and x is in quadrant IV.

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For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.

cos ( 2 θ ) = 3 5 and 90 θ 180

cos θ = 2 5 5 , sin θ = 5 5 , tan θ = 1 2 , csc θ = 5 , sec θ = 5 2 , cot θ = 2

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cos ( 2 θ ) = 1 2 and 180 θ 270

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For the following exercises, simplify to one trigonometric expression.

2 sin ( π 4 ) 2 cos ( π 4 )

2 sin ( π 2 )

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4 sin ( π 8 ) cos ( π 8 )

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For the following exercises, find the exact value using half-angle formulas.

sin ( π 8 )

2 2 2

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sin ( 11 π 12 )

2 3 2

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tan ( 3 π 8 )

1 2

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For the following exercises, find the exact values of a) sin ( x 2 ) , b) cos ( x 2 ) , and c) tan ( x 2 ) without solving for x .

If tan x = 4 3 , and x is in quadrant IV.

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If sin x = 12 13 , and x is in quadrant III.

a) 3 13 13 b) 2 13 13 c) 3 2

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If csc x = 7 , and x is in quadrant II.

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If sec x = 4 , and x is in quadrant II.

a) 10 4 b) 6 4 c) 15 3

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For the following exercises, use [link] to find the requested half and double angles.

Image of a right triangle. The base is length 12, and the height is length 5. The angle between the base and the height is 90 degrees, the angle between the base and the hypotenuse is theta, and the angle between the height and the hypotenuse is alpha degrees.

Find sin ( 2 θ ) , cos ( 2 θ ) , and tan ( 2 θ ).

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Find sin ( 2 α ) , cos ( 2 α ) , and tan ( 2 α ).

120 169 , 119 169 , 120 119

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Find sin ( θ 2 ) , cos ( θ 2 ) , and tan ( θ 2 ) .

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Find sin ( α 2 ) , cos ( α 2 ) , and tan ( α 2 ) .

2 13 13 , 3 13 13 , 2 3

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For the following exercises, simplify each expression. Do not evaluate.

cos 2 ( 28 ) sin 2 ( 28 )

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2 cos 2 ( 37 ) 1

cos ( 74 )

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1 2 sin 2 ( 17 )

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cos 2 ( 9 x ) sin 2 ( 9 x )

cos ( 18 x )

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4 sin ( 8 x ) cos ( 8 x )

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6 sin ( 5 x ) cos ( 5 x )

3 sin ( 10 x )

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For the following exercises, prove the identity given.

( sin t cos t ) 2 = 1 sin ( 2 t )

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sin ( 2 x ) = 2 sin ( x ) cos ( x )

2 sin ( x ) cos ( x ) = 2 ( sin ( x ) cos ( x ) ) = sin ( 2 x )

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cot x tan x = 2 cot ( 2 x )

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sin ( 2 θ ) 1 + cos ( 2 θ ) tan 2 θ = tan θ

sin ( 2 θ ) 1 + cos ( 2 θ ) tan 2 θ = 2 sin ( θ ) cos ( θ ) 1 + cos 2 θ sin 2 θ tan 2 θ = 2 sin ( θ ) cos ( θ ) 2 cos 2 θ tan 2 θ = sin ( θ ) cos θ tan 2 θ = cot ( θ ) tan 2 θ = tan θ

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For the following exercises, rewrite the expression with an exponent no higher than 1.

cos 2 ( 6 x )

1 + cos ( 12 x ) 2

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sin 4 ( 3 x )

3 + cos ( 12 x ) 4 cos ( 6 x ) 8

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cos 4 x sin 2 x

2 + cos ( 2 x ) 2 cos ( 4 x ) cos ( 6 x ) 32

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Technology

For the following exercises, reduce the equations to powers of one, and then check the answer graphically.

tan 4 x

3 + cos ( 4 x ) 4 cos ( 2 x ) 3 + cos ( 4 x ) + 4 cos ( 2 x )

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sin 2 x cos 2 x

1 cos ( 4 x ) 8

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tan 4 x cos 2 x

3 + cos ( 4 x ) 4 cos ( 2 x ) 4 ( cos ( 2 x ) + 1 )

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cos 2 ( 2 x ) sin x

( 1 + cos ( 4 x ) ) sin x 2

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tan 2 ( x 2 ) sin x

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For the following exercises, algebraically find an equivalent function, only in terms of sin x and/or cos x , and then check the answer by graphing both equations.

sin ( 4 x )

4 sin x cos x ( cos 2 x sin 2 x )

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Extensions

For the following exercises, prove the identities.

sin ( 2 x ) = 2 tan x 1 + tan 2 x

2 tan x 1 + tan 2 x = 2 sin x cos x 1 + sin 2 x cos 2 x = 2 sin x cos x cos 2 x + sin 2 x cos 2 x =
2 sin x cos x . cos 2 x 1 = 2 sin x cos x = sin ( 2 x )

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cos ( 2 α ) = 1 tan 2 α 1 + tan 2 α

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tan ( 2 x ) = 2 sin x cos x 2 cos 2 x 1

2 sin x cos x 2 cos 2 x 1 = sin ( 2 x ) cos ( 2 x ) = tan ( 2 x )

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( sin 2 x 1 ) 2 = cos ( 2 x ) + sin 4 x

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sin ( 3 x ) = 3 sin x cos 2 x sin 3 x

sin ( x + 2 x ) = sin x cos ( 2 x ) + sin ( 2 x ) cos x = sin x ( cos 2 x sin 2 x ) + 2 sin x cos x cos x = sin x cos 2 x sin 3 x + 2 sin x cos 2 x = 3 sin x cos 2 x sin 3 x

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cos ( 3 x ) = cos 3 x 3 sin 2 x cos x

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1 + cos ( 2 t ) sin ( 2 t ) cos t = 2 cos t 2 sin t 1

1 + cos ( 2 t ) sin ( 2 t ) cos t = 1 + 2 cos 2 t 1 2 sin t cos t cos t = 2 cos 2 t cos t ( 2 sin t 1 ) = 2 cos t 2 sin t 1

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sin ( 16 x ) = 16 sin x cos x cos ( 2 x ) cos ( 4 x ) cos ( 8 x )

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cos ( 16 x ) = ( cos 2 ( 4 x ) sin 2 ( 4 x ) sin ( 8 x ) ) ( cos 2 ( 4 x ) sin 2 ( 4 x ) + sin ( 8 x ) )

( cos 2 ( 4 x ) sin 2 ( 4 x ) sin ( 8 x ) ) ( cos 2 ( 4 x ) sin 2 ( 4 x ) + sin ( 8 x ) ) =                                                                                                    = ( cos ( 8 x ) sin ( 8 x ) ) ( cos ( 8 x ) + sin ( 8 x ) )                                                                                                    = cos 2 ( 8 x ) sin 2 ( 8 x )                                                                                                    = cos ( 16 x )

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Practice Key Terms 3

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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