Question

The number of principal solutions of $\tan 2\theta =1$ is

• A. One
• B. Two
• C. Three
• D. Four

Answer 1

Answer: (B)

Let $y=\tan 2\theta$
$\tan 2\theta =1$
$⟹2\theta ={\tan }^{-1}(1)$
$⟹2\theta =\frac{\pi }{4}+k\pi$, where $k$ is a positive integer
Now,
for $k=0\Rightarrow \theta =\frac{\pi }{4}$ and for $k=1\Rightarrow \theta =\frac{5\pi }{4}$
$0\le \frac{\pi }{4}\le 2\pi$ and $0\le \frac{5\pi }{4}\le 2\pi$
$\therefore 2\theta =\frac{\pi }{4},\frac{5\pi }{4}$
$⟹\theta =\frac{\pi }{8},\frac{5\pi }{8}$
Hence, the required principal solutions are $\frac{\pi }{8}$ and $\frac{5\pi }{8}$