Trigonometric Integral With Multiple Angles

by Shree

How do we integrate a function like this:

This type of problem is commonly found on most textbooks. It also appears in applications, for example in Fourier Series.

Your first reaction may be to try integration by parts, but in fact it is much simpler than that. We just use trigonometric identities. In this case, you need to remember the formulas for the sum and the difference of angles:

If we sum these two equations we get:

If we solve for the product in the right side:

This is a standard formula in trigonometry, but you can always do this little deduction whenever you need it (I always do that). So, we can apply this formula to our integral, making:

So, we get:

And this one is easy to calculate:

So, whenever you have an integral with product of sines or cosines with different angles, try to use the trig identities of sums and differences
of angles.

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