Trigonometrical Ratios And Identities Ques 12
- Given both $\theta$ and $\varphi$ are acute angles and $\sin \theta=\frac{1}{2}, \cos \varphi=\frac{1}{3}$, then the value of $\theta+\varphi$ belongs to
(2004, 1M)
(a) $\frac{\pi}{3}, \frac{\pi}{6}$
(b) $\frac{\pi}{2}, \frac{2 \pi}{3}$
(c) $\frac{2 \pi}{3}, \frac{5 \pi}{6}$
(d) $\frac{5 \pi}{6}, \pi$
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Answer:
Correct Answer: 12.(b)
Solution:
Formula:
- Since, $\sin \theta=\frac{1}{2}$
and $\cos \varphi=\frac{1}{3} \Rightarrow \quad \theta=\arccos\left(\frac{1}{3}\right)$ and $0<\cos \varphi=\frac{1}{3}<\frac{1}{2} \quad$ as $0<\frac{1}{3}<\frac{1}{2}$
$\Rightarrow \quad \theta=\frac{\pi}{6} \quad$ and $\quad \cos ^{-1}(0)>\varphi>\cos ^{-1} \quad \frac{1}{2}$
the sign changed as $\cos x$ is decreasing between $0$ and $\frac{\pi}{2}$
$$ \begin{array}{ll} \Rightarrow & \theta=\frac{\pi}{6} \quad \text { and } \quad \frac{\pi}{3}<\varphi<\frac{\pi}{2} \Rightarrow \frac{\pi}{2}<\theta+\varphi<\frac{5 \pi}{6} \\ \therefore & \theta \in \left( \frac{\pi}{2}, \frac{2 \pi}{3} \right) \end{array} $$
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