Question

Find the principal value $ta{n}^{-1}+co{s}^{-1}\left(\frac{-1}{2}\right)+si{n}^{-1}\left(\frac{-1}{2}\right)$

• A. $\frac{2\pi }{3}$
• B. $\frac{3\pi }{4}$
• C. $\frac{\pi }{2}$
• D. $6\pi$

Answer 1

Answer: (B)

Let $ta{n}^{-1}\left(1\right)=\theta$
$\Rightarrow tan\theta =1=tan\frac{\pi }{4}$
$\Rightarrow \theta =\frac{\pi }{4}\in \left(\frac{-\pi }{2},\frac{\pi }{2}\right)$
$\therefore$ Principal value of $ta{n}^{-1}\left(1\right)$ is $\frac{\pi }{4}$
Let $co{s}^{-1}\left(\frac{-1}{2}\right)=\varphi$
$\Rightarrow cos\varphi =\frac{-1}{2}$
$=-cos\frac{\pi }{3}$
$=cos\left(\pi -\frac{\pi }{3}\right)$
$=cos\frac{2\pi }{3}$
$\Rightarrow \varphi =\frac{2\pi }{3}\in \left[0,\pi \right]$
$\therefore$ Principal value of $co{s}^{-1}\left(\frac{-1}{2}\right)$ is $\frac{2\pi }{3}$
Also, principal value of $si{n}^{-1}\left(\frac{-1}{2}\right)$ is $\left(\frac{-\pi }{6}\right)$
$\therefore$ Principal value of $ta{n}^{-1}\left(1\right)+co{s}^{-1}\left(\frac{-1}{2}\right)+si{n}^{-1}\left(\frac{-1}{2}\right)$
$=\frac{\pi }{4}+\frac{2\pi }{3}-\frac{\pi }{6}$
$=\frac{3\pi }{4}$