# Question #14b3d

Dec 22, 2016

$\cot \left(2 \theta\right) + 5 \cos \theta = 2$

$\cos \frac{2 \theta}{\sin} \left(2 \theta\right) + 5 \cos \theta = 2$

$\frac{{\cos}^{2} \theta - {\sin}^{2} \theta}{2 \sin \theta \cos \theta} + 5 \cos \theta = 2$

$\frac{{\cos}^{2} \theta - {\sin}^{2} \theta + \left(5 \cos \theta\right) \cdot \left(2 \sin \theta \cos \theta\right)}{2 \sin \theta \cos \theta} = 2$

$\frac{{\cos}^{2} \theta - {\sin}^{2} \theta + 10 \sin \theta {\cos}^{2} \theta}{2 \sin \theta \cos \theta} = 2$

$\frac{1 - {\sin}^{2} \theta - {\sin}^{2} \theta + 10 \sin \theta \left(1 - {\sin}^{2} \theta\right)}{2 \sin \theta \cos \theta} = 2$

$\frac{1 - 2 {\sin}^{2} \theta + 10 \sin \theta - 10 {\sin}^{3} \theta}{2 \sin \theta \cos \theta} = 2$

$- 10 {\sin}^{3} \theta - 2 {\sin}^{2} \theta + 10 \sin \theta + 1 = 4 \sin \theta \cos \theta$

Using a graphing calculator, I graphed each side of the equation to find the points of intersection.
$\theta \approx 1.036 + 2 \pi n$
$\theta \approx 1.74 + 2 \pi n$
$\theta \approx 3.213 + 2 \pi n$