Prove the following orthogonality relations (which are used to...

Question

Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.

c.

π

∫ sin(mx) cos(nx) dx = 0, when |m + n| is even

0

Accepted Answer

Welcome back, everyone. What is the value of the integral from 0 toi of sine of 5 X multiplied by cosine of 3 X D X A 0 B 1, C 1/4, and D 12. For this problem we're going to convert our product into some. So let's recall the identity, sine alpha, cosine of beta. This product can be expressed as a 1/2 multiplied by. Sign of alpha plus beta plus sign of alpha minus beta. Our angle alpha is 5 X. And beta is 3x. Therefore, we can rewrite sine of 5 X, cosine of 3 X S 1/2 ranis. Sign of alpha plus beta, so that'd be 5 x + 3X, which is 8 X. Plus sin of alpha minus beta, so that'll be 5 X minus 3X, which is 2 X. Therefore we are integrating. From 0 to pi, a function, which is 1/2 we can factor out the constant in paris we get sine of 8 X plus sine of 2 X. And now we are ready to integrate. Let's recall that the integral of sin is cosine. So first of all, we get our constant, which is 1/2. In print C's integrating sin of 8 X, we get cosine of 8 X. And we have to divide it by the slope of the linear term, so that'll be 8. Plus, the integral of sine is cosine, so the integral of sine of 2 X is cosine of 2 x divided by the slope of the linear term. So that'll be 2. And now let's value the result from 0 to pi. So we got 1/2. In prints, we first will substitute 5 for every X, right? So we get cosine. Off ate pie. Divided by 8. Plus cosine of 2 pi. Divided by 2. And then what we want to do is simply subtract the value of this function at the lower limit, so we're subtracting 1/2. Empres Cosine of 8 multiplied by 0, which is cosine of 0 divided by 8. Plus cosine of 2 multiplied by 0, which is again cosine of 0. Divided by 2. And now let's simplify, so we get 1/2 in paris. Cosign of 8 i. Now, cosine of 8 by is going to be equal to 1, right? So we get 1 divided by 8 plus. What about cosine of 2 by? The value of cosine of 2 by is also one so we get 1/8 plus 1/2. And then we're subtracting one half ineries cosine of 0 divided by 8 cosine of 0 is 1, so we get 1/8. Plus cosine of 0 divided by 2, which is, once gain 1 divided by 2. So, cosine of 8, cosine of 2, and cosine of 0, all of these are equal to 1. And now let's notice that we're subtracting 1/2 multiplied by. The sum between 1/8 and 1/2 from itself, right? Meaning the answer to this problem is equal to 0, which corresponds to the answer choice a. Thank you for watching.

Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.
c.
π
∫ sin(mx) cos(nx) dx = 0, when |m + n| is even
0

Key Concepts

Orthogonality of Functions

Fourier Series

Properties of Sine and Cosine Functions

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