Transcript Chapter 5 Section 6

5
Trigonometric
Identities
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5.6-1
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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5.6-2
5.6 Half-Angle Identities
Half-Angle Identities ▪ Applying the Half-Angle Identities
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1.1-3
5.6-3
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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5.6-4
Half-Angle Identities
Choose the appropriate sign depending on the
quadrant of
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5.6-5
Half-Angle Identities
There are three alternative forms for
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5.6-6
Half-Angle Identities
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5.6-7
Double-Angle Identities
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1.1-8
5.6-8
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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1.1-9
5.6-9
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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1.1-10
5.6-10
Example 3
FINDING FUNCTION VALUES OF s/2
GIVEN INFORMATION ABOUT s
The angle associated with
is positive while
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lies in quadrant II since
are negative.
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5.6-11
Example 3
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FINDING FUNCTION VALUES OF s/2
GIVEN INFORMATION ABOUT s (cont.)
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5.6-12
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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1.1-13
5.6-13
Example 5
VERIFYING AN IDENTITY
Verify that
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is an identity.
1.1-14
5.6-14