Question

If $5cos2\theta +2co{s}^2\frac{\theta }{2}+1=\theta$, where $0<\theta <\mathrm{\pi }$ , then the value of $\theta$

• A. $\frac{\pi }{3}\pm \pi$
• B. $\frac{\pi }{3},{\cos }^{-1}\left(\frac{3}{5}\right)$
• C. ${\cos }^{-1}\left(\frac{3}{5}\right)\pm \pi$
• D. $\frac{\pi }{3},\pi –{\cos }^{-1}\left(\frac{3}{5}\right)$

Answer 1

The correct option is D

$\frac{\pi }{3},\pi –{\cos }^{-1}\left(\frac{3}{5}\right)$

Find the value of $\theta$:

Given, $5cos2\theta +2co{s}^2\frac{\theta }{2}+1=\theta$

$\Rightarrow$ $5(2\cos 2\theta -1)+1+\cos \theta +1=0$

$\Rightarrow$ $10\cos 2\theta -5+1+\cos \theta +1=0$

$\Rightarrow$ $10\cos 2\theta +\cos \theta -3=0$

$\Rightarrow$ $10\cos 2\theta +6\cos \theta –5\cos \theta -3=0$

$\Rightarrow$$2\cos \theta (5\cos \theta +3)–1(5\cos \theta +3)=0$

$\Rightarrow$ $(2\cos \theta –1)(5\cos \theta +3)=0$

$\Rightarrow$ $2\cos \theta –1=0,5\cos \theta +3=0$

$\Rightarrow$ $\cos \theta =\frac{1}{2},-\frac{3}{5}$

$\mathbf{\therefore }\mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{\pi }}{\mathbf{3}}\mathbf{,}\mathbf{}\mathit{\pi }\mathbf{}\mathbf{-}\mathbf{}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{-}\mathbf{1}}\left(\frac{3}{5}\right)$

Hence, option (D) is correct.