Integrating Products of Sines
and Cosines of Different
Angles
CK-12
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Printed: December 15, 2015
AUTHOR
CK-12
www.ck12.org
C HAPTER
Chapter 1. Integrating Products of Sines and Cosines of Different Angles
1
Integrating Products of
Sines and Cosines of Different Angles
Objectives
In this concept, you will learn how to compute the integrals of products of sine and cosine functions that have
different angles.
Concept
The previous concepts looked at integrals where products of same angle trig functions were used. What if the trig
function factors have different arguments (angles)? This is the case in many science and engineering problems. One
mathematical resource that is extremely useful for modeling some functions is the fourier series, which is an infinite
sum of sine waves. In simple terms, a function S(x) could be modeled as:
∞
S(x) = a1 sin x + a2 sin 2x + a3 sin 3x · · · =
∑ an sin nx.
n=1
The weights an reflect which frequencies are most prominent in the modeled function S(x). One of the key
Rπ
characteristics of the Fourier series is the “orthogonality” of each of the sine functions, defined as sin kx sin mxdx =
0
Rπ
0 for k 6= m. Can you evaluate the integral sin kx sin mxdx = 0 to show orthogonality?
0
Watch This
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/105722
http://www.youtube.com/watch?v=QdNScjd5bno - patrickJMT: Trigonometric Integrals, Part 5 (5:58)
Guidance
This Rsection looks at integrals
involving the product(s) of sine and cosine functions having different linear arguments,
R
e.g. sin(ax) sin(bx)dx, or sin2 (ax) cos(cx)dx. Use is made of the following three basic sine and cosine product
identities involving different arguments:
TABLE 1.1:
1
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TABLE 1.1: (continued)
Sine/Cosine Product Identity
sin A sin B = 21 [cos(A − B) − cos(A + B)]
Derived From
cos(A + B) = cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B
sin(A + B) = sin A cos B − cos A sin B
sin(A − B) = sin A cos B + cos A sin B
cos(A + B) = cos A cos B − sin A sin B
cos(A − B) = cos A cos B + sin A sin B
sin A cos B = 21 [sin(A − B) + sin(A + B)]
cos A cos B = 12 [cos(A − B) + cos(A + B)]
The procedure involves applying the identities to reduce the complexity of the integrand.
Example A
R
Evaluate sin x sin 3xdx
Solution:
1
[cos(x − 3x) − cos(x + 3x)] dx
2
Z
1
=
[cos(−2x) − cos(4x)] dx
2
1 sin 4x
1 sin(−2x)
−
=
2
−2
2
4
Z
sin(2x) sin 4x
sin x sin 3xdx =
−
+C
. . . Use
4
8
Z
Z
sin x sin 3xdx =
Example B
R
Evaluate sin(9x) cos(4x)dx
Solution:
Z
Z
sin(9x) cos(4x)dx =
1
[sin(9x − 4x) + sin(9x + 4x)] dx
2
1
[sin(5x) + sin(13x)] dx
2
1 − cos(5x)
1 − cos 13x
=
+
2
5
2
13
Z
cos(5x) cos(13x)
+
+C
sin(9x) cos(4x)dx = −
10
26
Z
=
Example C
R
Evaluate cos(9x) cos(5x)dx
Solution:
2
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Chapter 1. Integrating Products of Sines and Cosines of Different Angles
Z
Z
cos(9x) cos(5x)dx =
1
[cos(9x − 4x) + cos(9x + 4x)] dx
2
1
[cos(5x) + cos(13x)] dx
2
1 sin(5x)
1 sin 13x
=
+
+C
2
5
2
13
Z
sin(5x) sin 13x
sin(9x) cos(4x)dx =
+
+C
10
26
Z
=
Concept Question Wrap-up
Rπ
Can you evaluate the integral sin kx sin mxdx = 0 to show orthogonality?
0
Option 1: Using integration by parts (2 passes required) to solve for the integral yields
Zπ
sin kx sin mxdx =
1
m2 − k2
[k cos kπ sin mπ − m sin kπ cos mπ].
0
For the integral to be 0, m and k must be integers (sin mπ = sin kπ = 0)
Option 2: Using sum and difference angle properties for the integrand yields sin kx sin mx = 21 [cos(k − m)x − cos(k +
m)x]. The integral becomes
Zπ
sin kx sin mxdx =
1
[(k + m) sin(k − m)π − (k − m) sin(k + m)π].
2(k2 − m2 )
0
For the integral to be 0, (k − m) and (k + m) must be integers.
Guided Practice
Evaluate the integral sin2 2x cos 7xdx
R
Solution:
Z
1
(1 − cos 4x) cos 7xdx
2
Z
1
=
(cos 7x − cos 4x cos 7x)xdx
2
Z
Z
1
1
=
cos 7xdx −
cos 4x cos 7xdx
2
2
Z
1
1 1
=
sin 7x −
[cos 11x + cos 4x]dx
14
2 2
1
1 sin 11x sin 4x
=
sin 7x −
+
+C
14
4
11
4
1
1
1
=
sin 7x − sin 11x − sin 3x +C
14
44
12
sin2 2x cos 7xdx =
Z
3
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Practice
Evaluate the integrals:
1.
2.
3.
4.
5.
6.
R
R
R
R
R
R
7.
R2
8.
R2
sin 2x cos 3xdx
sin 4x sin 3xdx
cos 3x cos 4xdx
cos 2x cos 3xdx
cos 2x sin 3xdx
sin 2x sin 3xdx
π
0
4 sin x cos( 3x )dx
π
R0
4 sin( 3x ) cos(x)dx
9. R sin(6x) sin(2x)dx
10. R cos(2x) cos(−x)dx
11. R sin(10x) cos(5x)dx
12. sin3 (3x) cos(5x)dx
13.
R2
cos(3πx) cos2 (πx)dx
R0
14. R x sin 2x sin 3xdx
15. x cos 2x cos 3xdx
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 8.7.
4