Question

The angle between the curves $y=\sin x$ and $y=\cos x$ is

• A. ${\tan }^{-1}\left(2\sqrt{2}\right)$
• B. ${\tan }^{-1}\left(3\sqrt{2}\right)$
• C. ${\tan }^{-1}\left(3\sqrt{3}\right)$
• D. ${\tan }^{-1}\left(5\sqrt{2}\right)$

Answer 1

The correct option is A

${\tan }^{-1}\left(2\sqrt{2}\right)$

Explanation for the correct option:

Step 1: Finding the slope $m_1$ and $m_2$

Consider the given Equation

$y=\sin x$ and $y=\cos x$

$$\begin{gathered} \Rightarrow \sin x=\cos x \\ \Rightarrow \frac{\sin x}{\cos x}=1 \\ \Rightarrow \tan x=1 \end{gathered}$$

We know that from the trigonometric function

$x=45°\text{or}\frac{\mathrm{\pi }}{4}$

Differentiate the given Equation with respect to $x$

$$\begin{gathered} y=\sin x \\ \Rightarrow \frac{dy}{dx}=\cos x \\ \Rightarrow \frac{dy}{dx}=\cos \frac{\mathrm{\pi }}{4} \\ \Rightarrow m_1=\frac{1}{\sqrt{2}}\left[\because \cos \frac{\mathrm{\pi }}{4}=\frac{1}{\sqrt{2}}\right] \end{gathered}$$

And

$$\begin{gathered} y=\cos x \\ \Rightarrow \frac{dy}{dx}=-\sin x \\ \Rightarrow \frac{dy}{dx}=-\sin \frac{\mathrm{\pi }}{4} \\ \Rightarrow m_2=-\frac{1}{\sqrt{2}}\left[\because -\sin \frac{\mathrm{\pi }}{4}=-\frac{1}{\sqrt{2}}\right] \end{gathered}$$

Step 2: Finding the angle between the curves

We know that

$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$

Substituting the value of $m_1\text{and}{m}_2$

$$\begin{gathered} \tan \theta =\left|\frac{-{\displaystyle \frac{1}{\sqrt{2}}}-{\displaystyle \frac{1}{\sqrt{2}}}}{1+\left({\displaystyle \frac{1}{\sqrt{2}}}\right)\left(-{\displaystyle \frac{1}{\sqrt{2}}}\right)}\right| \\ \tan \theta =\left|\frac{{\displaystyle \frac{2}{\sqrt{2}}}}{1-{\displaystyle \frac{1}{2}}}\right| \\ \tan \theta =\left|\frac{{\displaystyle \frac{2}{\sqrt{2}}}}{{\displaystyle \frac{2-1}{2}}}\right| \\ \tan \theta =\frac{{\displaystyle \sqrt{2}\times \sqrt{2}}}{{\displaystyle \frac{\sqrt{2}}{{\displaystyle \frac{1}{2}}}}} \\ \tan \theta =2\sqrt{2} \\ \therefore \theta ={\tan }^{-1}2\sqrt{2} \end{gathered}$$

Hence, option (A) is the correct answer.