Transcript 5.5 Double Angle Identities

5
Trigonometric
Identities
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5 Trigonometric Identities
5.1 Fundamental Identities
5.2 Verifying Trigonometric Identities
5.3 Sum and Difference Identities for Cosine
5.4 Sum and Difference Identities for Sine
and Tangent
5.5 Double-Angle Identities
5.6 Half-Angle Identities
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5.5 Double-Angle Identities
Double-Angle Identities ▪ An Application ▪ Product-to-Sum and
Sum-to-Product Identities
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Double-Angle Identities
We can use the cosine sum identity to derive
double-angle identities for cosine.
Cosine sum identity
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Double-Angle Identities
There are two alternate forms of this identity.
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Double-Angle Identities
We can use the sine sum identity to derive a
double-angle identity for sine.
Sine sum identity
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Double-Angle Identities
We can use the tangent sum identity to derive a
double-angle identity for tangent.
Tangent sum identity
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Double-Angle Identities
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Example 1
FINDING FUNCTION VALUES OF 2θ
GIVEN INFORMATION ABOUT θ
Given
and sin θ < 0, find sin 2θ, cos 2θ, and tan
2θ.
Find Sin θ using the appropriate quadrant.
Now use the double-angle identity for sine.
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 4  3
sin2  2sin  cos   2       
 5  5
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Now find cos2θ, using the first double-angle identity for
cosine (any of the three forms may be used).
9 16
7
2
2
cos 2  cos   sin  


25 25
25
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Example 1
FINDING FUNCTION VALUES OF 2θ
GIVEN INFORMATION ABOUT θ (cont.)
Now find tan θ and then use the tangent doubleangle identity.
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Example 1
FINDING FUNCTION VALUES OF 2θ
GIVEN INFORMATION ABOUT θ (cont.)
Alternatively, find tan 2θ by finding the quotient of
sin 2θ and cos 2θ.
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sin2  25 24
tan 2 


cos 2  7
7
25
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Example 2
FINDING FUNCTION VALUES OF θ
GIVEN INFORMATION ABOUT 2θ
Find the values of the six trigonometric functions of θ if
We must obtain a trigonometric function value of θ
alone.
θ is in quadrant II, so sin θ is positive.
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Example 2
FINDING FUNCTION VALUES OF θ
GIVEN INFORMATION ABOUT 2θ (cont.)
Use a right triangle in quadrant II to find the values of
cos θ and tan θ.
Use the Pythagorean
theorem to find x.
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Example 3
Verify that
VERIFYING A DOUBLE-ANGLE IDENTITY
is an identity.
Quotient identity
Double-angle
identity
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Example 4
SIMPLIFYING EXPRESSIONS USING
DOUBLE-ANGLE IDENTITIES
Simplify each expression.
cos2A = cos2A – sin2A
Multiply by 1.
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Example 5
DERIVING A MULTIPLE-ANGLE
IDENTITY
Write sin 3x in terms of sin x.
Sine sum identity
Double-angle identities
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Product-to-Sum Identities
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Sum-to-Product Identities
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Half-Angle Identities
A
1  cos A
cos  
2
2
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