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

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

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

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

$2\cos \theta (\cos \theta -2)-(\cos \theta -2)=0$

$(2\cos \theta -1)(\cos \theta -2)=0$

$\cos \theta \ne 2$

$\therefore 2\cos \theta -1=0$

$\cos \theta =\frac{1}{2}=\cos \frac{\pi }{3}\therefore \theta =2n\pi \pm \frac{\pi }{3}$