Download Notes 5.5

5.5
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Multiple–Angle and
Product–to–Sum Formulas
Use multiple–angle formulas to rewrite and evaluate
trigonometric functions
Use power–reducing formulas to rewrite and evaluate
trigonometric functions
Use half–angle formulas to rewrite and evaluate
trigonometric functions
Use product–to–sum and sum–to–product formulas to
rewrite and evaluate trigonometric functions.
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Multiple–Angle Formulas
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Multiple–Angle Formulas
You should learn the double–angle formulas below because they are
used often in trigonometry and calculus.
Example 1: Solve:
2 cos x + sin 2x = 0.
2 cos x + 2 sin x cos x = 0
x=
2 cos x(1 + sin x) = 0
2 cos x = 0
cos x = 0
1 + sin x = 0
sin x = –1
+ 2n
and
x=
+ 2n
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Double and Triple Angle Formula
A.
C.
cos 2x
B.
tan 2x
Rewrite sin 4x in terms of sin x and cos x.
sin(2x + 2x)
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Verify using Double Angle Formulas
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Power–Reducing Formulas
The double–angle formulas can be used to obtain the following power–
reducing formulas.
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Reducing a Power
Example 4: Rewrite sin4 x as a sum of first powers of the cosines of
multiple angles.
=
(1 – 2 cos 2x + cos2 2x)
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Half–Angle Formulas
You can derive some useful alternative forms of the power–reducing
formulas by replacing u with u/2. The results are called half-angle formulas.
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Using a Half–Angle Formula
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Solving a Trigonometric Equation
Example 6: Find all solutions of
in the interval [0, 2).
cos x(cos x – 1) = 0
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Product–to–Sum Formulas
Each of the following product–to–sum formulas is easily verified using
the sum and difference formulas.
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Writing Products as Sums
Example 8:
Rewrite the product as a sum or difference.
cos 5x sin 4x
cos 5x sin 4x =
=
[sin(5x + 4x) – sin(5x – 4x)]
sin 9x –
sin x.
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Product–to–Sum Formulas
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Using a Sum–to–Product Formula
Example 9:
Find the exact value of cos 195° + cos 105°.
cos 195° + cos 105° =
= 2 cos 150° cos 45°
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Solving a Trig. Equation
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