TITLE: Angles between curves
QUESTION [0 upvotes]: Question: Find the angle between the curves $y = \sin 5x$ and $y = \cos 5x$ .
I have used the formula $ \tan\theta =\left| \dfrac{m_1-m_2}{1+m_1 m_2}  \right|    $
and found the gradient values by deriving each function.
I get an angle of 27.124 degrees. But the answer in the book says : 31.6 degrees ?

REPLY [0 votes]: Notice, the point of intersection of the curves $y=\sin 5x$ & $y=\cos 5x$ is found as $$\sin 5x=\cos 5x\implies \tan 5x=1$$ $$5x=\frac{\pi}{4}\ \ \ or  \ \ x=\frac{\pi}{20}$$
the slopes of the tangents at the point of intersection of curves, $x=\pi/20$,  are determined as follows 
$$m_1=\frac{d}{dx}(\sin 5x)=5\cos 5x=5\cos\frac{\pi}{4}=\frac{5}{\sqrt2}$$
$$m_2=\frac{d}{dx}(\cos 5x)=-5\sin 5x=5\sin\frac{\pi}{4}=-\frac{5}{\sqrt2}$$
hence, the angle between the curves $$\tan\theta=\tan^{-1}\left|\frac{\frac{5}{\sqrt2}-\left(-\frac{5}{\sqrt2}\right)}{1+\frac{5}{\sqrt2}\left(-\frac{5}{\sqrt2}\right)}\right|=\color{red}{\tan^{-1}\left(\frac{10\sqrt 2}{23}\right)\approx 31.5863^\circ}$$
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