Question

If $\cos \theta +sec\theta =\frac{5}{2}$, then the general value of $\theta$ is

• A. $n\pi \pm \frac{\pi }{3}$
• B. $2n\pi \pm \frac{\pi }{6}$
• C. $n\pi \pm \frac{\pi }{6}$
• D. $2n\pi \pm \frac{\pi }{3}$

Answer 1

The correct option is D

$2n\pi \pm \frac{\pi }{3}$

Explanation for the correct option:

Step 1. Find the value of $\theta$:

Given, $\cos \theta +sec\theta =\frac{5}{2}$

$\Rightarrow$ $\cos \theta +\frac{1}{\cos \theta }=\frac{5}{2}$

$\Rightarrow$ ${\cos }^2\theta +1=\frac{5}{2}\cos \theta$

$\Rightarrow$ $2{\cos }^2\theta +2=5\cos \theta$

$\Rightarrow$ $2{\cos }^2\theta –5\cos \theta +2=0$

$\Rightarrow$ $2{\cos }^2\theta –4\cos \theta –\cos \theta +2=0$

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

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

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

Step 2. $\cos \theta =2$ is not possible since range of $\cos \theta$ is $[-1,1]$.

$\Rightarrow$$\cos \theta =\frac{1}{2}$

$\Rightarrow$$\cos \theta =\cos \frac{\pi }{3}$

$\mathbf{\therefore }\mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}\mathit{n}\mathit{\pi }\mathbf{}\mathbf{\pm }\mathbf{}\frac{\mathbf{\pi }}{\mathbf{3}}$

Hence, Option ‘D’ is Correct.