Question

Find the principal and general solutions of the following equation:

$\sec x=2$

• A. Principal solution $=\frac{\pi }{6},\frac{5\pi }{6}$ and General solution $=n\pi \pm \frac{\pi }{6},n\in Z$
• B. Principal solution $=\frac{\pi }{3},\frac{5\pi }{3}$ and General solution $=n\pi \pm \frac{\pi }{3},n\in Z$
• C. Principal solution $=\frac{\pi }{6},\frac{5\pi }{6}$ and General solution $=2n\pi \pm \frac{\pi }{6},n\in Z$
• D. Principal solution $=\frac{\pi }{3},\frac{5\pi }{3}$ and General solution $=2n\pi \pm \frac{\pi }{3},n\in Z$

Answer 1

The correct option is D Principal solution $=\frac{\pi }{3},\frac{5\pi }{3}$ and General solution $=2n\pi \pm \frac{\pi }{3},n\in Z$

$secx=2$

$\Rightarrow \frac{1}{\cos x}=2$

$\Rightarrow \cos x=1/2$

we know that ${\cos 60}^0=1/2$

$cos$ is positive in $I$ & $IV$ quadrant. so, principal values $x=\frac{\pi }{3}$ and $x=\frac{5\pi }{3}$

For general solution-

let $\cos x=\cos y\to \left(1\right)$

given $\cos x=1/2\to \left(2\right)$

from eq $\left(1\right)$ & $\left(2\right)$

$\cos y=1/2$

$\Rightarrow \cos y=\cos \frac{\pi }{3}$

$\Rightarrow y=\frac{\pi }{3}$

$\because \cos x=\cos y$

General solution is

$x=2n\pi \pm y,n\in z$

put $y=\frac{\pi }{3}$

$\Rightarrow x=2n\pi \pm \frac{\pi }{3},n\in z$