Question

If $3(se{c}^2\theta +ta{n}^2\theta )=5$, then the general solution of $\theta$

• A. $2n\pi +\frac{\pi }{6}$
• B. $2n\pi \pm \frac{\pi }{6}$
• C. $n\pi \pm \frac{\pi }{6}$
• D. $n\pi \pm \frac{\pi }{3}$

Answer 1

Answer: (C)

$n\pi \pm \frac{\pi }{6}$

$3({\sec }^2\theta +{\tan }^2\theta )=5\Rightarrow {\tan }^2\theta +1+{\tan }^2\theta =\frac{5}{3}\Rightarrow 2{\tan }^2\theta +1=\frac{5}{3}$
$\Rightarrow {\tan }^2\theta =\frac{1}{3}={\left(\frac{1}{\sqrt{3}}\right)}^2={\left(\tan \frac{\pi }{6}\right)}^2$

$\Rightarrow \tan \theta =\pm \tan \frac{\pi }{6}$

$\Rightarrow \theta =n\pi \pm \frac{\pi }{6}$