Problem $2$:

$($a$)$ Using the basis $\{sin(t),sin(2t),sin(4t)\},$ defined for $t$ from $-1$ to $+1,$ find the best meansquare approximation of the function shown in Figure $2.$ Is this basis orthogonal?

$($b$)$ Find the best mean$-$square approximation of the function in Figure $2$ using the function $sin(2t)($i$.$e$.$ a basis with a single function$).$ Compare the errors of the two approximations.