Trigonometric Integrals

Trigonometric Integrals

Part 1: Some Prep Work

J. Gonzalez-Zugasti, University of Massachusetts - Lowell

1

 sin2 and cos2

Recall:

sin2

=

1

-

cos 2

2

=

1 2

-

1 4

sin

2

+

and

cos2

=

1

+

cos 2

2

=

1 2

+

1 4

sin

2

+

J. Gonzalez-Zugasti, University of Massachusetts - Lowell

2

Evaluate

Example 1

cos3

Solution: Use sin2 + cos2 = 1 cos2 = 1 - sin2 .

J. Gonzalez-Zugasti, University of Massachusetts - Lowell

3

Example 1 (continued)

cos3 = cos2 cos

= 1 - sin2 cos

= 1 - 2

= sin = cos

=

-

1 3

3

+

=

sin

-

1 3

sin3

+

J. Gonzalez-Zugasti, University of Massachusetts - Lowell

4

Evaluate

Example 2

sin4

Solution:

Use

sin2

=

1-cos 2

2

and

cos2

=

1+cos 2

2.

J. Gonzalez-Zugasti, University of Massachusetts - Lowell

5

Example 2 (continued)

sin4 = sin2 2

1 - cos 2 2

=

2

1 - 2cos 2 + cos2 2

=

4

=

1 4

1

-

2cos 2

+

cos2

2

J. Gonzalez-Zugasti, University of Massachusetts - Lowell

6

Example 2 (continued)

sin4

=

1 4

1

-

2cos 2

+

cos2

2

1

1 + cos 4

= 4 1 - 2cos 2 + 2

1 = 8 2 - 4cos 2 + 1 + cos 4

1 = 8 3 - 4cos 2 + cos 4

J. Gonzalez-Zugasti, University of Massachusetts - Lowell

7

Example 2 (continued)

sin4

=

1 8

3 - 4 cos 2 + cos 4

1

1

1

= 8 3 - 4 2 sin 2 + 4 sin 4 +

31

1

= 8 - 4 sin 2 + 32 sin 4 +

J. Gonzalez-Zugasti, University of Massachusetts - Lowell

8

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