# 5.6b Solving Trigonometric Equation (Factorization)

5.6b Solving Trigonometric Equation (Factorization)

Example:
Find all the angles that satisfy each of the following equations for £ x £ 360°.
(a) $\mathrm{cot}x=-2\mathrm{cos}x$
(b) $3\mathrm{sec}x=4\mathrm{cos}x$
(c) $16\mathrm{tan}x=\mathrm{cot}x$

Solution:
(a)
$\begin{array}{l}\mathrm{cot}x=-2\mathrm{cos}x\\ \frac{\mathrm{cos}x}{\mathrm{sin}x}=-2\mathrm{cos}x\\ \mathrm{cos}x=-2\mathrm{cos}x\mathrm{sin}x\\ \mathrm{cos}x+2\mathrm{sin}x\mathrm{cos}x=0\\ \mathrm{cos}x\left(1+2\mathrm{sin}x\right)=0\\ \mathrm{cos}x=0\\ x={90}^{\circ },{270}^{\circ }\\ 1+2\mathrm{sin}x=0\\ \mathrm{sin}x=-\frac{1}{2}\\ \text{Basic}\angle =\text{3}{0}^{\circ }\\ x=\left({180}^{\circ }+{30}^{\circ }\right),\left({360}^{\circ }-{30}^{\circ }\right)\\ x={210}^{\circ },{330}^{\circ }\\ \therefore x={90}^{\circ },{210}^{\circ },{270}^{\circ },{330}^{\circ }\end{array}$

(b)

(c)