Hprec9 3

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9-3: Other Identities Objectives: Use the double angle and half angle identities for sine, cosine & tangent ©2009 Roy L. Gover ( www.mrgover.com ) )

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Transcript of Hprec9 3

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9-3: Other IdentitiesObjectives:•Use the double angle and half angle identities for sine, cosine & tangent

©2009 Roy L. Gover (www.mrgover.com)

)

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Analysis

Does sin 90º =2 sin 45°?

Does cos 30°= cos 60°?

1

2

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Analysis

sin( )

sin cos cos sin

From Lesson 9-2:

Let , then… x

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Important Idea

sin 2 2sin cosx x x

Double Angle Identity for sine:

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Example

If sin x = 4/5 and x is in the first quadrant, find the exact value of sin 2x.…first, find cos x…

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Try This

If sin x = 3/5 and x is in the first quadrant, find the exact value of sin 2x

24sin 2

25x

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Try ThisIf sin x = 3/5 and x is in the second quadrant, find the exact value of sin 2x. 24

sin 225

x

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Analysis

cos( )

cos cos sin sin

From Lesson 9-2:

Let , then… x

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Important Idea

2 2cos 2 cos sin x x x

Double Angle Identity for cosine:

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Double Angle Identity for cos

Three forms...

1. cos 2x = cos2x - sin2x

2. cos 2x = 2cos2x - 1

3. cos 2x = 1 - 2sin2x

Important Idea

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Example

If sin x is 1/5 and x is in the first quadrant, find the exact value of cos 2x.

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Try ThisIf sin x is 2/5 and x is in the second quadrant, find the exact value of cos 2x.

17cos 2

25x

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What would you do to derive a double angle identity for tangent?

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Double Angle Identity for Tangent:

2

2 tantan 2

1 tan

x

xx

Important Idea

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Try This

If , and is in the first quadrant, find .

5cos

5x x

tan 2x

Hint: first find …

tan x

4tan 2

3x

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Analysis

2cos 2 2cos 1

1 cos 2cos

2

One of the double angle identities:

Solve for cos :

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Analysis

1 cos 2os

2c

Let 22

x

x

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Analysis

1 coscos

2 2

x x

which results in:

The half angle identity for cosine.

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Important Idea

1 cossin

2 2

x x

1 coscos

2 2

x x

1 costan

2 1 cos

x x

x

The half angle identities: See

page 596 of your text.

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Example Use a half angle identity to find the exact value of sin 22°30’Hint: Multiply the given angle by 2 then divide by 2

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Example Use a half angle identity to find the exact value of19

cos12

Hint: Multiply the given angle by 2 then divide by 2

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Try This Use a half angle identity to find the exact value of cos 105°

210 2 3cos105 cos

2 2

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Try This

Use a half angle identity

to find the exact value of

55 2 24sin sin8 2 2

5sin

8