This document discusses multiple angle identities in trigonometry. It covers double angle identities, power-reducing identities, half-angle identities, and examples of using identities to solve trigonometric equations and find heights of triangles. Specific identities covered include sin2x, cos2x, tan2x, and expressions for reducing powers of trig functions like sin4x to single powers.

1. 5.4
Multiple
Angle
Identities
Copyright © 2011 Pearson, Inc.

2. What you’ll learn about
 Double-Angle Identities
 Power-Reducing Identities
 Half-Angle Identities
 Solving Trigonometric Equations
… and why
These identities are useful in calculus courses.
Copyright © 2011 Pearson, Inc. Slide 5.4 - 2

3. Double Angle Identities
sin2u  2sinu cosu
cos2u 
cos2 u  sin2 u
2cos2 u 1
1 2sin2 u





tan2u 
2 tanu
1 tan2 u
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4. Proving a Double-Angle Identity
cos2x  cos(x  x)
 cos x cos x - sin x sin x
 cos2 x  sin2 x
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5. Power-Reducing Identities
sin2 u 
1 cos2u
2
cos2 u 
1 cos2u
2
tan2 u 
1 cos2u
1 cos2u
Copyright © 2011 Pearson, Inc. Slide 5.4 - 5

6. Example Reducing a Power of 4
Rewrite sin4 x in terms of trigonometric functions with
no power greater than 1.
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7. Example Reducing a Power of 4
sin4 x  sin2 x2

1 cos2x
2

 

 
2

1 2cos2x  cos2 2x
4

1
4

cos2x
2

1
4
1 cos 4x
2

 

 

1
4

cos2x
2

1 cos 4x
8
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8. Half-Angle Identities
sin
u
2
 
1 cosu
2
cos
u
2
 
1 cosu
2
tan
u
2


1 cosu
1 cosu
1 cosu
sinu
sinu
1 cosu









Copyright © 2011 Pearson, Inc. Slide 5.4 - 8

9. Example Using a Double Angle
Identity
Solve cos x  cos3x  0 in the interval [0,2 ).
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10. Example Using a Double Angle
Identity
Solve cos x  cos3x  0 in the interval [0,2 ).
Solve Graphically
The graph suggest that
there are six solutions:
0.79, 1.57, 2.36,
3.93, 4.71, 5.50.
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11. Example Using a Double Angle
Identity
Solve cos x  cos3x  0 in the interval [0,2 ).
Confirm Algebraically
cos x  cos3x  0
cos x  cos x cos2x  sin xsin2x  0
cos x  cos x1 2sin2 x sin x2sin x cos x 0
cos x  cos x  2cos x sin2 x  2cos x sin2 x  0
2cos x  4 cos x sin2 x  0
2cos x 1 2sin2  x 0
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12. Example Using a Double Angle
Identity
2cos x1 2sin2 x 0
cos x  0 or 1 2sin2 x  0
x 

2
or
3
2
or sin x  
2
2
x 

2
or
3
2
or x 

4
,
3
4
,
5
4
, or
7
4
The six exact solutions in the given interval are


3
5
3
7
,
,
,
,
, and
.
4
2
4
4
2
4
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13. Quick Review
Find the general solution of the equation.
1. cot x 1  0
2. (sin x)(1 cos x)  0
3. cos x  sin x  0
4. 2sin x  22sin x 1 0
5. Find the height of the isosceles triangle with
base length 6 and leg length 4.
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14. Quick Review Solutions
Find the general solution of the equation.
1. cot x 1  0 x 
3
4
  n
2. (sin x)(1 cos x)  0 x   n
3. cos x  sin x  0 x 

4
  n
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15. Quick Review Solutions
Find the general solution of the equation.
4. 2sin x  22sin x 1 0
x 
5
4
 2 n, x 
7
4
 2 n,
x 

6
 2 n, x 
5
6
 2 n
5. Find the height of the isosceles triangle with
base length 6 and leg length 4. 7
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