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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

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
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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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