2.3.4 So­lu­tion dot-d

Ques­tion:

Show that the func­tions $\sin(x)$ and $\cos(x)$ are or­thog­o­nal on the in­ter­val 0 $\raisebox{-.3pt}{$\leqslant$}$ $x$ $\raisebox{-.3pt}{$\leqslant$}$ $2\pi$.

An­swer:

They are by de­f­i­n­i­tion or­thog­o­nal if the in­ner prod­uct is zero. Check that:

$$\left\langle\vphantom{\cos(x)}\sin(x)\hspace{-\nulldelimiter... ...(2x)\bigg\vert _0^{2\pi} = \frac 14 \Big(1-\cos(4\pi)\Big) = 0$$