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

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For the following exercise, simplify the expression.

tan ( 1 2 x ) + tan ( 1 8 x ) 1 tan ( 1 8 x ) tan ( 1 2 x )

tan ( 5 8 x )

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For the following exercises, find the exact value.

cos ( sin 1 ( 0 ) cos 1 ( 1 2 ) )

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tan ( sin 1 ( 0 ) + sin 1 ( 1 2 ) )

3 3

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Double-Angle, Half-Angle, and Reduction Formulas

For the following exercises, find the exact value.

Find sin ( 2 θ ) , cos ( 2 θ ) , and tan ( 2 θ ) given cos θ = 1 3 and θ is in the interval [ π 2 , π ] .

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Find sin ( 2 θ ) , cos ( 2 θ ) , and tan ( 2 θ ) given sec θ = 5 3 and θ is in the interval [ π 2 , π ] .

24 25 , 7 25 , 24 7

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sec ( 3 π 8 )

2 ( 2 + 2 )

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For the following exercises, use [link] to find the desired quantities.

Image of a right triangle. The base is 24, the height is unknown, and the hypotenuse is 25. The angle opposite the base is labeled alpha, and the remaining acute angle is labeled beta.

sin ( 2 β ) , cos ( 2 β ) , tan ( 2 β ) , sin ( 2 α ) , cos ( 2 α ) ,  and  tan ( 2 α )

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sin ( β 2 ) , cos ( β 2 ) , tan ( β 2 ) , sin ( α 2 ) , cos ( α 2 ) ,  and  tan ( α 2 )

2 10 , 7 2 10 , 1 7 , 3 5 , 4 5 , 3 4

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For the following exercises, prove the identity.

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

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

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

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For the following exercises, rewrite the expression with no powers.

cos 2 x sin 4 ( 2 x )

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tan 2 x sin 3 x

10 sin x 5 sin ( 3 x ) + sin ( 5 x ) 8 ( cos ( 2 x ) + 1 )

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Sum-to-Product and Product-to-Sum Formulas

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

cos ( π 3 ) sin ( π 4 )

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2 sin ( 2 π 3 ) sin ( 5 π 6 )

3 2

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2 cos ( π 5 ) cos ( π 3 )

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For the following exercises, evaluate the sum by using a product formula. Write the exact answer.

sin ( π 12 ) sin ( 7 π 12 )

2 2

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cos ( 5 π 12 ) + cos ( 7 π 12 )

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For the following exercises, change the functions from a product to a sum or a sum to a product.

sin ( 9 x ) cos ( 3 x )

1 2 ( sin ( 6 x ) + sin ( 12 x ) )

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cos ( 7 x ) cos ( 12 x )

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sin ( 11 x ) + sin ( 2 x )

2 sin ( 13 2 x ) cos ( 9 2 x )

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cos ( 6 x ) + cos ( 5 x )

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Solving Trigonometric Equations

For the following exercises, find all exact solutions on the interval [ 0 , 2 π ) .

tan x + 1 = 0

3 π 4 , 7 π 4

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2 sin ( 2 x ) + 2 = 0

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For the following exercises, find all exact solutions on the interval [ 0 , 2 π ) .

2 sin 2 x sin x = 0

0 , π 6 , 5 π 6 , π

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cos 2 x cos x 1 = 0

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2 sin 2 x + 5 sin x + 3 = 0

3 π 2

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cos x 5 sin ( 2 x ) = 0

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1 sec 2 x + 2 + sin 2 x + 4 cos 2 x = 0

No solution

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For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval [ 0 , 2 π ) . Round to four decimal places.

3 cot 2 x + cot x = 1

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csc 2 x 3 csc x 4 = 0

0.2527 , 2.8889 , 4.7124

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For the following exercises, graph each side of the equation to find the zeroes on the interval [ 0 , 2 π ) .

20 cos 2 x + 21 cos x + 1 = 0

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sec 2 x 2 sec x = 15

1.3694 ,   1.9106 ,   4.3726 ,   4.9137

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Modeling with Trigonometric Equations

For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table.

x 0 1 2 3 4 5
y 1 6 11 6 1 6
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x y
0 2
1 1
2 2
3 5
4 2
5 1

3 sin ( x π 2 ) 2

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x y
3 3 + 2 2
2 3
1 2 2 1
0 1
1 3 2 2
2 1
3 −1 2 2
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A man with his eye level 6 feet above the ground is standing 3 feet away from the base of a 15-foot vertical ladder. If he looks to the top of the ladder, at what angle above horizontal is he looking?

71.6

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Using the ladder from the previous exercise, if a 6-foot-tall construction worker standing at the top of the ladder looks down at the feet of the man standing at the bottom, what angle from the horizontal is he looking?

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Practice Key Terms 2

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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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