How to easily visualise $ \int_{-\pi}^{\pi} \cos (nx)\cos(kx)dx=0$?

Question

Is there an easy way to visualise that the following holds?

$$ \int_{-\pi}^{\pi} \cos (nx)\cos(kx)dx=0$$

I thought about simply switching the cosine to sine:

$$ \int_{-\pi}^{\pi} \cos (nx)\sin\left( kx+\frac{\pi}{2}\right)dx=0$$

Now its an even function times an odd function which makes an odd function. This will integrate over a symmetric interval to zero, which is very easy to visualise. However I have the feeling this is mistaken because you could also do it for $\cos^2 (nx) $.

I tend to avoid simply trigonometric substitutions because they are very hard to visualise.

I tried using kronecker method of integration, but this doesn't work:

$$ \int \cos (nx)\cos(kx)dx = \cos(nx)\sin(kx)-\sin(nx)\cos(kx)+...$$

EDIT: I think I found a solution:

Rewrite:

$$ \int_{-\pi}^{\pi} \cos (nx)\cos(kx)dx$$

$$ \int_{-\pi}^{\pi} \cos (nx)\cos(nx+mx)dx$$

Where $k=n+m$

Now, whenever $mx=\frac{1}{2}\pi$ we get:

$$ \int_{-\pi}^{\pi} \cos (nx)\cos(nx+\frac{1}{2}\pi)dx$$ Which can be rewritten as:

$$ \int_{-\pi}^{\pi} \cos (nx)\sin(nx)dx=0$$

Which is zero because an even function ($\cos$) times an odd function ($\sin$) is odd. And on a symmetric interval the integral of an odd function is zero.

Here is a visualizer although the integral likely is $0$ only when $n\ne k;n,k\in\Bbb Z$ — Тyma Gaidash, Dec 09 '22 at 13:44
Thank you, I mean a more intuitive visualisation or a more simple way of understanding that it must be zero. There must be an easy way of seeing it. — bananenheld, Dec 09 '22 at 14:05
I was trying to find a different solution than yours, because it is just plugging in a trigonometric substitution without really seeing what is going on. — bananenheld, Jan 23 '23 at 20:22

score 0 · Accepted Answer · answered Jan 23 '23 at 20:20

The Kronecker method actually works, i just forgot to evaluate them

$$ \int_{-π}^{+π} \cos (nx)\cos(kx)dx = c_1\cos(nx)\sin(kx)|_{-π}^{+π}+c_2 \sin(nx)\cos(kx)|_{-π}^{+π}+c_3... $$

Here $c_i$ are constants related to integration/differentiation (e.g. $c_1=1/k$)

$$\cos(nx)\sin(kx)|_{-π}^{+π}=0, \ \ \ \ \ k\in \mathbf{N}$$ $$ \int_{-π}^{+π} \cos (nx)\cos(kx)dx = 0+0+0+0+\dots$$ $$ \boxed{\int_{-π}^{+π} \cos (nx)\cos(kx)dx = 0}$$

How to easily visualise $ \int_{-\pi}^{\pi} \cos (nx)\cos(kx)dx=0$?

1 Answers1