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Trigonometrical Equations
easy
If $2{\sin ^2}\theta = 3\cos \theta ,$ where $0 \le \theta \le 2\pi $, then $\theta = $
A
$\frac{\pi }{6},\frac{{7\pi }}{6}$
B
$\frac{\pi }{3},\frac{{5\pi }}{3}$
C
$\frac{\pi }{3},\frac{{7\pi }}{3}$
D
None of these
(IIT-1963)
Solution
(b) $2 – 2{\cos ^2}\theta = 3\cos \theta $
==> $2{\cos ^2} + 3\cos \theta – 2 = 0$
==> $\cos \theta = \frac{{ – 3 \pm \sqrt {9 + 16} }}{4} = \frac{{ – 3 \pm 5}}{4}$
Neglecting $(-)$ sign, we get
$\cos \theta = \frac{1}{2} = \cos \left( {\frac{\pi }{3}} \right)$
$ \Rightarrow $ $\theta = 2n\pi \pm \frac{\pi }{3}$.
The values of $\theta $ between $0$ and $2\pi $ are $\frac{\pi }{3},{\rm{ }}\frac{{{\rm{5}}\pi }}{{\rm{3}}}$.
Standard 11
Mathematics
