Chapter 5 Analytic Trigonometry
Section 5.5 Multiple-Angle and Product-to-Sum Formulas
Course/Section
Lesson Number
Date
Section Objectives: Students will know how to use multiple-angle formulas,
power-reducing formulas, half-angle formulas, and
product-to-sum formulas to rewrite and evaluate
trigonometric functions.
I. Multiple-Angle Formulas (pp. 407−409)
State the following double-angle formulas.
sin 2u = 2sin u cos u
cos 2u = cos2 u – sin2 u
= 2cos2 u – 1
= 1 – 2sin2 u
2tan u
tan 2u
1 tan 2 u
Pace: 15 minutes
Example 1. Solve sin 2x – cos x = 0.
sin 2x – cos x = 0
2sin x cos x – cos x = 0
cos x (2sin x – 1) = 0
cos x = 0 or sin x = 1/2
x = /2, 3 /2 or x = /6, 5 /6
Example 2. Sketch the graph of y = cos4 x – sin4 x on [0, 2 ].
y = cos4 x – sin4 x = (cos2 x – sin2 x) (cos2 x + sin2 x)
= cos 2x 1 = cos 2x
Example 3. Given sin x = 12/13 and /2 < x < , find sin 2x, cos 2x,
and tan 2x.
cos x = −5/13
sin 2x = 2sin x cos x = 2(12/13)( −5/13) = −120/169
cos 2x = 1 – 2sin2 x = 1 – 2(12/13)2 = −119/169
sin 2 x 120
tan 2x
cos 2x 119
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5.5-1
II. Power-Reducing Formulas (p. 409)
State the following power-reducing formulas.
1 cos 2u
1 cos2u
sin 2 u
cos 2 u
tan 2 u
2
2
Pace: 5 minutes
1 cos2u
1 cos2u
Example 4. Rewrite cos4 x as a sum of first powers of the cosine
function.
cos 4 x
cos 2 x
2
2
1 cos 2x
2
1
1 2 cos2x cos2 2x
4
1
1 cos 4 x
1 2cos 2x
4
2
1
3 4cos 2x cos 4x
8
III. Half-Angle Formulas (pp. 410−411)
State the following half-angle formulas.
u
1 cosu
sin
2
2
u
2
u
tan
2
cos
Pace: 10 minutes
1 cosu
2
1 cos u
sin u
sin u
1 cos u
Example 5. Find the exact value of cos 165 .
1 cos 330o
2
cos165o
1
3 2
2
2
3
2
Example 6. Solve 2sin2 (x/2) = cos x on [0, 2 ).
x
2sin 2
cos x
2
2
2
1 cos x
2
2
cos x
1 cos x
2
cos x
1 cos x
1
cos x
x
5.5-2
cos x
2cos x
1
2
5
,
3
3
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IV. Product-to-Sum Formulas (pp. 411−413)
State the following product-to-sum formulas.
1
sin u sin v
cos u v cos u v
2
1
cos u cos v
cos u v cos u v
2
1
sin u cos v
sin u v sin u v
2
1
cosu sin v
sin u v sin u v
2
Pace: 15 minutes
Example 7. Rewrite sin 2x sin x as a difference.
1
cos x
sin 2 x sin x
cos 2 x x cos 2x x
2
2
cos3 x
2
State the following sum-to-product formulas.
x y
x y
cos
sin x sin y 2 sin
2
2
sin x sin y
2 cos
cos x
2 cos
cos x
cos y
cos y
x
y
2
x
2sin
y
2
x
y
2
x
sin
y
2
x
cos
y
2
sin
x
y
2
Example 8. Find the exact value of sin 195 − sin 105 .
195o 105o
195o 105o
sin
sin195o sin105o 2 cos
2
2
2 cos150o sin 45o
2
2
3
2
2
6
2
Example 9. Solve cos 3x + cos x = 0 on [0, 2 ).
cos3x cos x 0
2 cos
3x
x
2
x
2
2 cos 2x cos x
cos
3x
cos2x
2x
x
cos x
0
0
0
3
5
7
,
,
2
2
2
3
5
7
,
,
,
4
4
4
4
3
0 x
,
2
2
2
,
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5.5-3
Example 10. Verify the identity.
sin 7x sin 5x
cos 7x cos 5x
cot 6x
7x 5x
7 x 5x
sin
2
2
7x 5x
7x 5 x
sin
2 sin
2
2
cos6 x
sin 6x
cot 6x
2 cos
sin 7x sin 5x
cos 7x cos 5x
V. Application (p. 414)
Pace: 10 minutes
• Assign the Writing About Mathematics on page 414 of the text.
5.5-4
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