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 0° £ x £ 360°.

(a)

$\cot x = - 2 \cos x$

(b)

$3 \sec x = 4 \cos x$
(c)

$16 \tan x = \cot x$

Solution:
(a)

$\begin{array}{l} \cot x = - 2 \cos x \\ \frac{\cos x}{\sin x} = - 2 \cos x \\ \cos x = - 2 \cos x \sin x \\ \cos x + 2 \sin x \cos x = 0 \\ \cos x (1 + 2 \sin x) = 0 \\ \cos x = 0 \\ x = 90^{\circ}, 270^{\circ} \\ 1 + 2 \sin x = 0 \\ \sin x = - \frac{1}{2} \\ Basic ∠ = 3 0^{\circ} \\ x = (180^{\circ} + 30^{\circ}), (360^{\circ} - 30^{\circ}) \\ x = 210^{\circ}, 330^{\circ} \\ ∴ x = 90^{\circ}, 210^{\circ}, 270^{\circ}, 330^{\circ} \end{array}$

(b)

$\begin{array}{l} 3 \sec x = 4 \cos x \\ \frac{3}{\cos x} = 4 \cos x \\ 3 = 4 \cos^{2} x \\ \cos^{2} x = \frac{3}{4} \\ \cos x = \pm \frac{\sqrt{3}}{2} \\ Basic ∠ = 30^{\circ} \\ x = 30^{\circ}, (180^{\circ} - 30^{\circ}), (180^{\circ} + 30^{\circ}), (360^{\circ} - 30^{\circ}) \\ x = 30^{\circ}, 150^{\circ}, 210^{\circ}, 330^{\circ} \end{array}$

(c)

$\begin{array}{l} 16 \tan x = \cot x \\ 16 \tan x = \frac{1}{\tan x} \\ \tan^{2} x = \frac{1}{16} \\ \tan x = \pm \frac{1}{4} \\ Basic ∠ = {14.04}^{\circ} \\ x = {14.04}^{\circ}, (180^{\circ} - {14.04}^{\circ}), \\ (180^{\circ} + {14.04}^{\circ}), (360^{\circ} - {14.04}^{\circ}) \\ x = {14.04}^{\circ}, {165.96}^{\circ}, {194.04}^{\circ}, {345.96}^{\circ} \end{array}$

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