Question

Solve the equation and write general solution: 2 sin² θ - 4 = 5 cos θ.

Answer 1

To solve the equation 2 sin² θ - 4 = 5 cos θ

we must convert it into in sin or in cos

by using the formula
${sin}^{2} \theta = 1 - {cos}^{2}\theta$
$2(1 - {cos}^{2} \theta) - 4 - 5 \: cos \: \theta = 0 \\ \\ 2 - 4 - 2 {cos}^{2}\theta - 5 \: cos \: \theta = 0 \\ \\ 2 {cos}^{2}\theta + 5 \: cos \: \theta + 2= 0 \\ \\ 2 {cos}^{2}\theta + 4\: cos \: \theta + cos \: \theta + 2= 0 \\ \\ 2 \: cos \: \theta(cos \: \theta + 2) + 1(cos \: \theta + 2) = 0 \\ \\ (cos \: \theta + 2)(2 \: cos \: \theta + 1) = 0$
So

$cos \: \theta + 2 = 0 \\ \\ cos \: \theta = - 2$
Discard this value of θ because cos cannot reach till -2.

$2cos \: \theta + 1 = 0 \\ \\ cos \: \theta = \frac{ - 1}{2} \\ \\ \theta= {cos}^{ - 1} ( \frac{ - 1}{2} ) \\ \\ \theta = {cos}^{ - 1}(cos( \frac{\pi}{3} )$
since
$cos \: ( - \theta) = cos \: \theta$
Solution of equation:

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

General Solution: Hence
the general solution is

$\theta = ±\frac{\pi}{3}+2\:k\:\pi$

where k is any integer