# Solve:4cosθ-3secθ=2tanθ?

Jun 22, 2018

$x = {5}^{\circ} 74 + k {360}^{\circ}$
$x = {174}^{\circ} 26 + k {360}^{\circ}$
$x = {216}^{\circ} 87 + k {360}^{\circ}$
$x = {323}^{\circ} 13 + k {360}^{\circ}$)

#### Explanation:

$4 \cos t - \frac{3}{\cos t} = \frac{2 \sin t}{\cos t}$
Multiply both sides by cos t:
$4 {\cos}^{2} t - 3 = 2 \sin t$
$4 - 4 {\sin}^{2} t - 3 = 2 \sin t$
$4 {\sin}^{2} t + 2 \sin t - 1 = 0$
$D = {d}^{2} = {b}^{2} - 4 a c = 4 + 4 = 8$ --> $d = \pm 2 \sqrt{2}$
There are 2 real roots:
$\sin x = - \frac{b}{2 a} \pm \frac{d}{2 a} = - \frac{2}{8} \pm \frac{2 \sqrt{2}}{8} = \frac{- 1 \pm \sqrt{2}}{4}$
$\sin x = \frac{- 1 + \sqrt{2}}{4} = 0.10$
$\sin x = \frac{- 1 - \sqrt{2}}{4} = - 0.60$
a. sin x = 0.10
Calculator and unit circle give 2 solutions for x:
$x = {5}^{\circ} 74$, and $x = 180 - 5.74 = {174}^{\circ} 26$
b. sin x = -0.60
Calculator and unit circle -->
$x = - {36}^{\circ} 87$, and $x = 180 - \left(- 36.87\right) = {216}^{\circ} 87$
Note x = - 36.87 is co-terminal to $x = {323}^{\circ} 13$
For general answers, add $k {360}^{\circ}$