Question

The principal solution(s) for $\cos x=-\frac{1}{\sqrt{2}}$ is/are

• A. $\frac{3\pi }{4}$
• B. $\frac{5\pi }{4}$
• C. $\frac{11\pi }{4}$
• D. $\frac{13\pi }{4}$

Answer 1

Answer: (A), (B)

$\frac{5\pi }{4}$

$\cos x=-\frac{1}{\sqrt{2}}$
$\Rightarrow \cos x=\cos (\pi -\frac{\pi }{4})$
$\Rightarrow \cos x=\cos \left(\frac{3\pi }{4}\right)$
$\Rightarrow x=2n\pi \pm \frac{3\pi }{4},n\in Z...\left(1\right)$

We know that, for principal solutions, $\theta \in [0,2\pi ]$
Now, put $n=0$ in $\left(1\right)$, we get $x=\frac{3\pi }{4}$
Similarly, put $n=1$, we get $x=2\pi -\frac{3\pi }{4}=\frac{5\pi }{4}$
$\therefore$ Principal solutions are $\frac{3\pi }{4},\frac{5\pi }{4}.$