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 $\alpha_1 = \cos x$?
 A: 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. 
A: This is an orthogonal basis in the function space.
You need to find the coordinates $\{\alpha_{i}\}$ of the function $f(x) = (\cos{x}+\sin{x})^3$ on the basis $\{u_{i}(x)\}$.
To do this, you must to calculate the scalar products on the interval $[-\pi, \pi]$:
$$
\alpha_{i} = \frac{1}{\pi} \int\limits_{-\pi}^{\pi}{f(x) \cdot u_{i} (x) ~ dx}
$$
It's same as Fourier series!
http://en.wikipedia.org/wiki/Dot_product#Functions
http://en.wikipedia.org/wiki/Fourier_series#Hilbert_space_interpretation
A: Use $$\sin^3x=\frac{\sin 3x-3\sin x }{4},\cos^3 x=(1-\sin^2x)\cos x$$
and $$\sin 2x=2\sin x\cos x; \cos 2x=2\cos^2x-1=1-2\sin^2x=\cos^2x-\sin^2x$$
A: Have you tried the power reduction formulas, stated in this wikipedia article?
see, http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formula
