Download Trigonometric Identities - Phoenix Union High School District

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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5.6 Half-Angle Identities
Half-Angle Identities ▪ Applying the Half-Angle Identities ▪
Verifying an Identity
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Half-Angle Identities
We can use the cosine sum identities to derive halfangle identities.
Choose the appropriate sign depending on the
quadrant of
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Half-Angle Identities
1  cos 2 x
cos x 
2
1  cos2 x
cos x  
2
A
1  cos A
cos  
2
2
Choose the appropriate sign depending on the
quadrant of
2
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Half-Angle Identities
There are three alternative forms for
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Half-Angle Identities
A
sin A
tan 
,
2 1  cos A
From the identity
an equivalent identity.
we can also derive
A 1  cos A
tan 
2
sin A
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Half-Angle Identities
A
1  cos A
cos  
2
2
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Example 1
USING A HALF-ANGLE IDENTITY TO
FIND AN EXACT VALUE
Find the exact value of cos 15° using the half-angle
identity for cosine.
Choose the positive
square root.
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Example 2
USING A HALF-ANGLE IDENTITY TO
FIND AN EXACT VALUE
Find the exact value of tan 22.5° using the identity
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Example 3
FINDING FUNCTION VALUES OF s/2
GIVEN INFORMATION ABOUT s
The angle associated with
lies in quadrant II since
is positive while
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are negative.
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Example 3
FINDING FUNCTION VALUES OF s/2
GIVEN INFORMATION ABOUT s (cont.)
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Example 4
SIMPLIFYING EXPRESSIONS USING
THE HALF-ANGLE IDENTITIES
Simplify each expression.
This matches part of the identity for
Substitute 12x for A:
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Example 5
Verify that
VERIFYING AN IDENTITY
is an identity.
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