Find the coordinates of the expression $(\cos x + \sin x)^3$ in the basis {$1, \sin x, \cos x, \sin 2x, \cos 2x, \sin3x, \cos 3x$} I'm quite stumped here. I expanded out $(\cos x + \sin x)^3$ to $\sin^3x + \cos^3x + 3\sin x\cos^2x + 3\sin^2x\cos x$ And I've tried the trig identities for $\sin 3x = 3\sin x - 4\sin^3x$
I can't have a variable in the coordinates of the basis right? It has to be of some $\alpha_i$ that is a scalar to the linear combination; for example, I can't do something as $\alpha_1 = \cos x$?

When you expand, you get $\cos^3 x+3\cos^2 x\sin x+3\cos x\sin^2 x+\sin^3 x$.
Note that $3\cos^2 x\sin x=3(1-\sin^2 x)\sin x=3\sin x-3\sin^3 x$, and similarly 
$3\cos  x\sin^2 x=3\cos x-3\cos^3 x$. 
So our expression is equal to $3\cos x+3\sin x-2\cos^3 x-2\sin^3 x$. 
You know how to express $\sin^3 x$ in terms of $\sin 3x$ and $\sin x$. For from your expression we have $\sin^3 x=\frac{3\sin x-\sin 3x}{4}$. 
Similarly, you can use $\cos 3x=4\cos^3 x-3\cos x$ to express $\cos^3 x$ as a linear combination of $\cos x$ and $\cos 3x$.
The functions $1$, $\cos 2x$, and $\sin 2x$ do not enter the game. Or if you want them to, they enter with coordinates $0$. 
Remark: About your question, the coordinates have to be real numbers, not functions. 
