Question

The principal solution(s) for $\tan x=-1$ is/are

• A. $x=\frac{3\pi }{4}$
• B. $x=\frac{11\pi }{4}$
• C. $x=\frac{7\pi }{4}$
• D. $x=-\frac{\pi }{4}$

Answer 1

Answer: (A), (C)

The correct options are
A $x=\frac{3\pi }{4}$
C $x=\frac{7\pi }{4}$

$\tan x=-1=\tan (\pi -\frac{\pi }{4})$
$\Rightarrow \tan x=\tan \frac{3\pi }{4}$
$\Rightarrow x=n\pi +\frac{3\pi }{4},n\in Z\Rightarrow x=\frac{3\pi }{4},\frac{7\pi }{4},\frac{11\pi }{4},\dots$

We know that, for principal solutions, $\theta \in [0,2\pi ]$
Hence, the principal solutions are
$\frac{3\pi }{4},\frac{7\pi }{4}$