notes 4-4
DESCRIPTION
Basic Identities Involving Sines, Cosines, and TangentsTRANSCRIPT
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Section 4-4Basic Identities Involving Sines, Cosines, and
Tangents
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Identity
An equation that is true for all possible values of the variable
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Example 1Complete the following in your calculator.
cos2 30° + sin2 30° sin2 3π4
⎛⎝⎜
⎞⎠⎟+ cos2 3π
4⎛⎝⎜
⎞⎠⎟
sin2 −25°( ) + cos2 −25°( ) cos2 4π( ) + sin2 4π( )
1 1
1 1
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Pythagorean Identity
For all theta,
cos2 θ( ) + sin2 θ( ) = 1
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Example 2 If sinθ = 1
3, find cosθ.
sin2θ + cos2θ = 1
13( )2
+ cos2θ = 1
− 1
9 − 1
9
cos2θ = 8
9
cos2θ = ± 8
9
cosθ = ± 8
3
cosθ = ± 2 2
3
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Opposites TheoremFor all theta,
cos −θ( ) = cosθ
sin −θ( ) = − sinθ
tan −θ( ) = − tanθ
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Example 3
a.cos30° =
32
. Find cos −30°( ) b.sin −
π4
⎛⎝⎜
⎞⎠⎟= −
22
. Find − sinπ4
⎛⎝⎜
⎞⎠⎟
32
−2
2
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Supplements TheoremFor all theta in radians,
sin π −θ( ) = sinθ
cos π −θ( ) = − cosθ
tan π −θ( ) = − tanθ
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Complements TheoremFor all theta in radians,
sinπ2−θ
⎛⎝⎜
⎞⎠⎟= cosθ
cosπ2−θ
⎛⎝⎜
⎞⎠⎟= sinθ
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Example 4 If sin x = .681, find sin -x( ) and sin π - x( ).
sin -x( ) = −.681
sin π − x( ) = .681
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Half-turn TheoremFor all theta in radians,
cos π +θ( ) = − cosθ
sin π +θ( ) = − sinθ
tan π +θ( ) = tanθ
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Example 5Using the unit circle, explain why for all .
sin π −θ( ) = sinθ θ
On the unit circle, . When you measure theta, you start at . So, you’re beginning at points that are
reflections of each other. As you plot the values, you will notice they remain as reflections over the y-axis, which will keep the y-coordinates the same, which is .
π = 180° 0°
sinθ
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Homework
p. 256 #1 - 24