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QuestionMathsClass 11

If $$sin\theta =\frac{1}{2}$$, $$cos\phi =\frac{1}{3}$$, then $$\theta +\phi$$ belongs to, where $$0\lt \theta , \phi \lt \frac{\pi }{2}$$（ ）
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. $$\left(\frac{5\pi }{6},\pi \right)$$

B
4.6
4.6

Solution

Given:$$\sin\theta=\frac{1}{2},\cos\phi=\frac{1}{3}$$ and $$\theta\in(0,\frac{\pi}{2})$$.
$$\implies \theta=\frac{\pi}{6}\cdots(i)$$
Since,$$0\lt\frac{1}{3}\lt\frac{1}{2}$$
We have $$\frac{\pi}{3}\lt \phi\lt\frac{\pi}{2}$$
Adding $$\theta$$ on each sides of the inequality, we get
$$\frac{\pi}{3}+\theta\lt \phi+\theta\lt\frac{\pi}{2}+\theta$$
From (i) $$\frac{\pi}{3}+\frac{\pi}{6}\lt \phi+\theta\lt\frac{\pi}{2}+\frac{\pi}{6}$$
$$\implies (\theta+\phi )\in (\frac{\pi}{2},\frac{2\pi}{3})$$.
Hence, correct option is B