# Question #f1faf

Apr 23, 2017

$\textcolor{red}{2.625}$

#### Explanation:

$\sec 292.5 = \sec \left(\frac{585}{2}\right) = \frac{1}{\cos} \left(\frac{585}{2}\right)$

Let us calculate $\cos \left(\frac{585}{2}\right)$

$\cos \left(\frac{585}{2}\right) = \cos \left[\frac{720 - 135}{2}\right] = \cos \left(360 - \frac{135}{2}\right)$

$= \cos \left(- \frac{135}{2}\right) = \cos \left(\frac{135}{2}\right)$. ----------(1.)

Now,
$\cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}}$

$\therefore \cos \left(\frac{135}{2}\right) = \pm \sqrt{\frac{1 + \cos 135}{2}}$

$\cos 135 = \cos \left(180 - 45\right) = - \cos 45 = - \frac{1}{\sqrt{2}} = - 0.71$

$\implies \cos \left(\frac{135}{2}\right) = \pm \sqrt{\frac{1 - 0.71}{2}}$

$= \pm \sqrt{\frac{0.29}{2}} = \pm \sqrt{.0145} = \pm 0.381$

but $\frac{135}{2} = 67.5 < 90.$ Hence $\cos \left(\frac{135}{2}\right)$ will be positive.

$\therefore \cos \left(\frac{135}{2}\right) = 0.381$

From (1.) $\cos \left(\frac{585}{2}\right) = \cos \left(\frac{135}{2}\right) = 0.381$

$\implies \sec 292.5 = \sec \left(\frac{585}{2}\right) = \frac{1}{\cos} \left(\frac{585}{2}\right) = \frac{1}{.381} = \textcolor{red}{2.625}$

If you need help with the half angle formula, do comment below.