Find the [[../Area between Curves|Area between Curves]] between $f(x)=3\sin x$ and $g(x)=4\cos x$ in the range $\ba{0, 0.9\pi}$ $\huge \begin{align} f(0) &< g(0) \\ 3\sin x - 4 \cos x &= 0 \\ \frac{3}{4}\sin x &= \cos x \\ \\ \frac{3\tan x}{4} &= 1\\ \tan x &= \frac{4}{3} \\ \let r &= \arctan \frac{4}{3} \\ \\ A &=\int_{0}^{r} \pa{4 \cos x - 3\sin x}\d x + \int_{r}^{0.9\pi} \pa{3\sin x-4\cos x}\d x \\ &= \pa{-4\sin r -3\cos r} - \pa{ -3 } + \pa{3\cos 0.9\pi + 4\sin 0.9\pi } - \pa{3\cos r + 4\sin r} \\ &= 3 -4\sin r -3\cos r + 3\cos 0.9\pi + 4\sin 0.9\pi - 3\cos r - 4\sin r \\ &= 3 -8\sin r -6 \cos r+3\cos 0.9\pi + 4\sin 0.9\pi \\ &= 3 -8\sin (\arctan (\frac{4}{3} )) -6 \cos (\arctan (\frac{4}{3}))+3\cos 0.9\pi + 4\sin 0.9\pi \end{align} $ $\huge \begin{align} \end{align}$