8 4 relationships among the functions
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8.4 Relationships Among the Functions
Objectives:
1. Simplify trig expressions
2. Prove trig identities
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Relationships with Negatives
If P is a point onthe unit circle, then x = cos θand y = sin θ.
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Pythagorean Identities
Since P(cos θ, sin θ) is on the unit circle:
We’ll verify these later!
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To Simplify:
Use trig identities to try to express as one trig function or as a constant.
It often helps to rewrite everything in terms of sine and cosine.
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Example:
Simplify the expression.
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Example:
Simplify the expression.
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Example:
Simplify:
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You Try!
Simplify each expression:
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Investigating Cofunctions
Use your calculator to evaluate:
sin 50° cos 40°
sin 25° cos 65°
sin 79° cos 11°
sin 7° cos 83°What do you notice?
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Why Does This Happen?
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Cofunction Relationships
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Proving Identities
Simplify one (or both) side(s) of the equation to show they are equal.
Need to work with each side separately. Don’t move anything across the = sign.
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Example:
Prove:
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Example:
Prove:
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You Try!
Prove: