Angles Between Sine And Cosine

The question basically asks for angle between the curves at the point of intersection of the 2 curves

Solving the 2 curves, $y=\sin x$ and $y= \cos x$

$x=\dfrac{ \pi }{4}$

Let the slopes of $y=\sin x$ and $y= \cos x$ at $x=\dfrac{ \pi}{4}$ be $m_1$ and $m_2$ respectively then,

$m_1=\dfrac{d( \sin x)}{dx}= \cos x = \dfrac{1}{\sqrt{2}}$

$m_2=\dfrac{d( \cos x)}{dx}=- \sin x = - \dfrac{1}{\sqrt{2}}$

Acute angle between 2 lines whose slopes are $m_1$ and $m_2$ is given by,

$\tan \theta = \Biggl \lvert \dfrac{m_1 - m_2}{1+m_1m_2} \Biggr \rvert$

On substituting the values of $m_1$ and $m_2$ ,

$\tan \theta = \dfrac{ \frac{1}{ \sqrt{2}} - \frac{-1}{ \sqrt{2}}}{1+ \frac{1}{ \sqrt{2}} . \frac{-1}{ \sqrt{2}}}$

$\tan \theta = \dfrac{ \frac{2}{ \sqrt2 }}{ \frac{2-1}{2}}$

$\tan \theta = 2 \sqrt2$

$\color{#20A900}{ \boxed{ \therefore \theta = \tan ^{-1}(2 \sqrt{2})}}$