Question

If ${\theta }_1$ and ${\theta }_2$ are acute angle satisfying $\sin {\theta }_1=\frac{1}{2},\cos {\theta }_2=\frac{1}{3}$, then $({\theta }_2+{\theta }_1)$ lies in between

• A. $\left[\frac{\pi }{3},\frac{\pi }{2}\right]$
• B. $\left[\frac{\pi }{2},\frac{2\pi }{3}\right]$
• C. $\left[\frac{2\pi }{3},\frac{5\pi }{6}\right]$
• D. $\left[\frac{5\pi }{6},\pi \right]$

Answer 1

The correct option is B $\left[\frac{\pi }{2},\frac{2\pi }{3}\right]$

$\sin {\theta }_1=\frac{1}{2}$

$\cos {\theta }_2=\frac{1}{3}$

${\theta }_1$ and ${\theta }_2$ both are acute angles.

${\theta }_1={\sin }^{-1}\left(\frac{1}{2}\right)$

$\therefore$ ${\theta }_1=\frac{\pi }{6}$

$\Rightarrow$ $0<\frac{1}{3}<\frac{1}{2}$

$\Rightarrow$ $\cos \frac{\pi }{2}<\cos {\theta }_2<\cos \frac{\pi }{3}$

$\Rightarrow$ or $\frac{\pi }{3}<{\theta }_2<\frac{\pi }{2}$

$\therefore$ $\frac{\pi }{3}+\frac{\pi }{6}<{\theta }_1+{\theta }_2<\frac{\pi }{2}+\frac{\pi }{6}$

$\Rightarrow$ Or $\frac{\pi }{2}<{\theta }_1+{\theta }_2<\frac{2\pi }{3}$

$\Rightarrow$ ${\theta }_1+{\theta }_2\in [\frac{\pi }{2},\frac{2\pi }{3}]$