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Given sin α = 5 8 , with θ in quadrant I, find cos ( 2 α ) .

cos ( 2 α ) = 7 32

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Using the double-angle formula for cosine without exact values

Use the double-angle formula for cosine to write cos ( 6 x ) in terms of cos ( 3 x ) .

cos ( 6 x ) = cos ( 3 x + 3 x ) = cos 3 x cos 3 x sin 3 x sin 3 x = cos 2 3 x sin 2 3 x
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Using double-angle formulas to verify identities

Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.

Using the double-angle formulas to verify an identity

Verify the following identity using double-angle formulas:

1 + sin ( 2 θ ) = ( sin θ + cos θ ) 2

We will work on the right side of the equal sign and rewrite the expression until it matches the left side.

( sin θ + cos θ ) 2 = sin 2 θ + 2 sin θ cos θ + cos 2 θ = ( sin 2 θ + cos 2 θ ) + 2 sin θ cos θ = 1 + 2 sin θ cos θ = 1 + sin ( 2 θ )
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Verify the identity: cos 4 θ sin 4 θ = cos ( 2 θ ) .

cos 4 θ sin 4 θ = ( cos 2 θ + sin 2 θ ) ( cos 2 θ sin 2 θ ) = cos ( 2 θ )

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Verifying a double-angle identity for tangent

Verify the identity:

tan ( 2 θ ) = 2 cot θ tan θ

In this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.

tan ( 2 θ ) = 2 tan θ 1 tan 2 θ Double-angle formula = 2 tan θ ( 1 tan θ ) ( 1 tan 2 θ ) ( 1 tan θ ) Multiply by a term that results in desired numerator . = 2 1 tan θ tan 2 θ tan θ = 2 cot θ tan θ Use reciprocal identity for   1 tan θ .
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Verify the identity: cos ( 2 θ ) cos θ = cos 3 θ cos θ sin 2 θ .

cos ( 2 θ ) cos θ = ( cos 2 θ sin 2 θ ) cos θ = cos 3 θ cos θ sin 2 θ

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Use reduction formulas to simplify an expression

The double-angle formulas can be used to derive the reduction formulas    , which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses, calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from the double-angle formulas.

We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Let’s begin with cos ( 2 θ ) = 1 2 sin 2 θ . Solve for sin 2 θ :

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

Next, we use the formula cos ( 2 θ ) = 2 cos 2 θ 1. Solve for cos 2 θ :

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

The last reduction formula is derived by writing tangent in terms of sine and cosine:

tan 2 θ = sin 2 θ cos 2 θ = 1 cos ( 2 θ ) 2 1 + cos ( 2 θ ) 2 Substitute the reduction formulas . = ( 1 cos ( 2 θ ) 2 ) ( 2 1 + cos ( 2 θ ) ) = 1 cos ( 2 θ ) 1 + cos ( 2 θ )

Reduction formulas

The reduction formulas    are summarized as follows:

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

Writing an equivalent expression not containing powers greater than 1

Write an equivalent expression for cos 4 x that does not involve any powers of sine or cosine greater than 1.

We will apply the reduction formula for cosine twice.

cos 4 x = ( cos 2 x ) 2 = ( 1 + cos ( 2 x ) 2 ) 2 Substitute reduction formula for cos 2 x . = 1 4 ( 1 + 2 cos ( 2 x ) + cos 2 ( 2 x ) ) = 1 4 + 1 2 cos ( 2 x ) + 1 4 ( 1 + cos 2 ( 2 x ) 2 ) Substitute reduction formula for cos 2 x . = 1 4 + 1 2 cos ( 2 x ) + 1 8 + 1 8 cos ( 4 x ) = 3 8 + 1 2 cos ( 2 x ) + 1 8 cos ( 4 x )
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Questions & Answers

answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
what is a algebra
Jallah Reply
(x+x)3=?
Narad
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
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Nokwanda
C'est comment
Abdel
Hi
Amanda
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SORIE
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Chinni
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ANSHU
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Chinni
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SORIE
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SORIE
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
Practice Key Terms 3

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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