# Cos²α+cos²β+cos²γ = 1

1. Mar 19, 2012

### RK7

On pg 6 of http://www.scribd.com/doc/3914281/Crystal-Structure, it quotes this result without proof. My notes from uni also quote this result but I can't see where it comes from. Does anyone know? Thanks

Last edited: Mar 19, 2012
2. Mar 19, 2012

### mathman

Re: cos²α+cos²β+cos²γ=1

It looks like the Pythagorean theorem applied in three dimensions.

3. Mar 19, 2012

### HallsofIvy

Re: cos²α+cos²β+cos²γ=1

Well, it isn't true for just any angle $\alpha$, $\beta$, $\gamma$, of course! If, however, you draw a line through the origin of a three dimensional coordinate system, define $\alpha$ to be the angle the line makes with the x-axis, $\beta$ to be the angle the line makes with the y-axis, and $\gamma$ to be the angle the line makes with the z-axis, then this is true.

To see that, think of a vector of unit length in the direction of that line. If we drop a perpendicular from the tip of the vector to the x-axis, we have a right triangle in which an angle is $\alpha$ and the hypotenuse is 1, the length of the vector. Thus, the projection of the vector on the x-axis, and so the x-component of the vector is $cos(\alpha)$. Similarly, the y-component of the vector is $cos(\beta)$ and the z-component of the vector is $cos(\gamma)$. That is, $cos^2(\alpha)+ cos^2(\beta)+ cos^2(\gamma)$ is the square of the length of the vector which is, of course, 1.

By the way, look what happens if you do this in two dimensions. If you have a line in the plane through the origin making angles $\alpha$ with the x-axis and $\beta$ with the y- axis then $\beta= \pi/2- \alpha$ so $cos^2(\alpha)+ cos^2(\beta)= cos^2(\alpha)+ cos^2(\pi/2- \alpha)= cos^2(\alpha)+ sin^2(\alpha)= 1$.

4. Mar 19, 2012

### RK7

Re: cos²α+cos²β+cos²γ=1

That was embarrassing.. thanks