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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 establish an identity

Establish 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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Establish 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

what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
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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