1. Double-Angle and Half-Angle Identities
SECTION 6.3
Use double-angle identities.
Use power-reducing identities.
Use half-angle identities. 1 2 3

2. DOUBLE-ANGLE IDENTITIES
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EXAMPLE 1
Using Double-Angle Identities
If and  is in quadrant II, find the exact value of each expression.
Solution
First, we use identities to find sin θ and tan θ.
θ is in QII so sin > 0.
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EXAMPLE 1
Using Double-Angle Identities
Solution continued
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EXAMPLE 1
Using Double-Angle Identities
Solution continued
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7. Using the Double-Angle Formula for Tangent to Find an Exact Value
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Verifying an Identity
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10. Verifying an Identity (continued)
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EXAMPLE 3
Finding a Triple-Angle Identity for Sines
Verify the identity sin 3x = 3 sin x – 4 sin3 x.
Solution
sin 3x = sin (2x + x)
= sin 2x cos x + cos 2x sin x
= (2 sin x cos x) cos x + (1 – 2 sin2 x) sin x
= 2 sin x cos2 x + sin x – 2 sin3 x
= 2 sin x (1 – sin2 x) + sin x – 2 sin3 x
= 2 sin x – 2 sin3 x + sin x – 2 sin3 x
= 3 sin x – 4 sin3 x
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12. POWER REDUCING IDENTITIES
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EXAMPLE 4
Using Power-Reducing Identities
Write an equivalent expression for cos4 x that contains only first powers of cosines of multiple angles.
Solution
Use power-reducing identities repeatedly.
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EXAMPLE 4
Using Power-Reducing Identities
Solution continued
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15. HALF-ANGLE IDENTITIES
The sign, + or –, depends on the quadrant in which lies.
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EXAMPLE 6
Using Half-Angle Identities
Use a half-angle formula to find the exact value of cos 157.5º.
Solution
Because 157.5º = , use the half-angle identity for cos with θ = 315°. Because
lies in quadrant II, cos is negative.
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EXAMPLE 6
Using Half-Angle Identities
Solution continued
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18. Verifying an Identity

19. Half-Angle Formulas for Tangent
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Verifying an Identity
We worked with the right side and arrived at the left side. Thus, the identity is verified.
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