 # 6.6b Solving Trigonometric Equation (Factorization)

6.6b Solving Trigonometric Equation (Factorization)

Example:
Find all the angles that satisfy each of the following equations for £ £ 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)
$\begin{array}{l}3\mathrm{sec}x=4\mathrm{cos}x\\ \frac{3}{\mathrm{cos}x}=4\mathrm{cos}x\\ 3=4{\mathrm{cos}}^{2}x\\ {\mathrm{cos}}^{2}x=\frac{3}{4}\\ \mathrm{cos}x=±\frac{\sqrt{3}}{2}\\ \text{Basic}\angle ={30}^{\circ }\\ x={30}^{\circ },\left({180}^{\circ }-{30}^{\circ }\right),\left({180}^{\circ }+{30}^{\circ }\right),\left({360}^{\circ }-{30}^{\circ }\right)\\ x={30}^{\circ },{150}^{\circ },{210}^{\circ },{330}^{\circ }\end{array}$

(c)
$\begin{array}{l}16\mathrm{tan}x=\mathrm{cot}x\\ 16\mathrm{tan}x=\frac{1}{\mathrm{tan}x}\\ {\mathrm{tan}}^{2}x=\frac{1}{16}\\ \mathrm{tan}x=±\frac{1}{4}\\ \text{Basic}\angle \text{}={14.04}^{\circ }\\ x={14.04}^{\circ },\left({180}^{\circ }-{14.04}^{\circ }\right),\\ \text{}\left({180}^{\circ }+{14.04}^{\circ }\right),\left({360}^{\circ }-{14.04}^{\circ }\right)\\ x={14.04}^{\circ },{165.96}^{\circ },{194.04}^{\circ },{345.96}^{\circ }\end{array}$