Question

The most general value of $\theta$ which satisfies both the equations $\tan \theta =\sqrt{3}and\mathrm{cosec}\theta =-\frac{2}{\sqrt{3}}$ is-

• A. $n\pi +\frac{4\pi }{3}:n\in I$
• B. $n\pi \frac{2\pi }{3}:n\in I$
• C. $\in 2n\pi +\frac{4\pi }{3}:n\in I$
• D. $2n\pi +\frac{2\pi }{3}:n\in I$

Answer 1

Answer: (C)

$\in 2n\pi +\frac{4\pi }{3}:n\in I$

$\tan \theta =\sqrt{3}$

$\tan \theta =\tan \frac{\pi }{3}$

$\theta =\frac{\pi }{3},\frac{4\pi }{3},..............$

$cosec\theta =-\frac{2}{\sqrt{3}}$

$\theta =\frac{4\pi }{3}$ $[\theta =\frac{4\pi }{3}$satisfy the negative value of $cosec\theta =-\frac{2}{\sqrt{3}}]$

$\because \theta =2n\pi +\frac{4\pi }{3}$

When $n=1$

$\theta =2\pi +\frac{4\pi }{3}$

$\theta =3\pi +\frac{\pi }{3}$

$\tan (3\pi +\frac{\pi }{3})=\sqrt{3}$

$cosec(3\pi +\frac{\pi }{3})=-\frac{2}{\sqrt{3}}$

For $\theta =2n\pi +\frac{4\pi }{3}$ satisfies for values of $\tan \theta$ and $cosec\theta .$