# 6.6c Solving Trigonometric Equation (Form Quadratic Equation in sinx/ cosx/ tanx/ cosecx/ secx/ cotx)

(C) Solving Trigonometric Equation (Form Quadratic Equation in sinx/ cosx/ tanx/ cosecx/ secx/ cotx)

Example:
Find all the angles between 0° and 360° that satisfy each of the following equations.
(a) 3 sin² x – 2 sin x – 1 = 0
(b)  2 sin x = cosec x + 1
(c) 5 sin² x = 2 (1 + cos x)
(d) 2 sec x = 1 + cos x
(e)  2 cot² x + 8 = 7 cosec x

Solution:
(a)
3 sin² x – 2 sin x – 1 = 0
(3 sin x + 1)(sin x – 1) = 0
sin x = –, sin x = 1
sin x = –
basic angle = 19.47°
x = 180° + 19.47°, 360° – 19.47°
x = 199.47°, 340.53
sin x = 1, x = 90°
Hence x = 90°, 199.47°, 340.53°

(b)

2 sin x = cosec x + 1
$2\mathrm{sin}x=\frac{1}{\mathrm{sin}x}+1$
2 sin ² x = 1 + sin x
2 sin ² x – sin x – 1 =0
(2 sin x + 1)(sin x – 1) = 0
sin x = –½, sin x = 1
sin x = –½
basic angle = 30°
x = 180° + 30°, 360° – 30°
x = 210°, 330°
sin x = 1, x = 90°
Hence x = 90°, 210°, 330°

(c)

5 sin² x = 2 (1 + cos x)
5 (1 – cos² x) = 2 + 2 cos x
5 – 5 cos² x – 2 – 2 cos x = 0
– 5 cos² x – 2 cos x + 3 = 0
5 cos² x + 2 cos x – 3 = 0
(5 cos x – 3)(cos x + 1) = 0
$\begin{array}{l}\mathrm{cos}x=\frac{3}{5},\text{}\mathrm{cos}x=-1\\ \mathrm{cos}x=\frac{3}{5}\end{array}$
basic angle = 53.13°
x = 53.13°, 360° – 53.13°
x = 53.13°, 306.87°
cos x = – 1
x = 180°
Hence x = 53.13°, 180°, 306.87°

(d)

2 sec x = 1 + cos x
$\frac{2}{\mathrm{cos}x}=1+\mathrm{cos}x$
2 = cos x + cos² x
cos² x + cos x – 2 = 0
(cos x – 1)(cos x + 2) = 0
cos x = 1
x = 0°, 360°
cos x = –2 (not accepted)
Hence x = 0°, 360°

(e)
2 cot² x + 8 = 7 cosec x
2 (cosec² x – 1) + 8 = 7 cosec x
2 cosec² x – 2 – 7 cosec x + 8 = 0
2 cosec2x – 7 cosec x + 6 = 0
(2 cosec x – 3)(cosec x – 2) = 0
$\begin{array}{l}\mathrm{cos}ecx=\frac{3}{2},\text{}\mathrm{cos}ecx=2\\ \mathrm{sin}x=\frac{2}{3},\text{}\mathrm{sin}x=\frac{1}{2}\\ \mathrm{sin}x=\frac{2}{3}\\ \text{basic angle =}{41.81}^{\circ }\\ x={41.81}^{\circ },\text{}180-{41.81}^{\circ }\\ \mathrm{sin}x=\frac{1}{2}\\ {\text{basic angle = 30}}^{\circ }\\ x={30}^{\circ },\text{}180-{30}^{\circ }\\ \text{Hence}x\text{}=\text{}{30}^{\circ },\text{}{41.81}^{\circ },\text{}{138.19}^{\circ },\text{}{150}^{\circ }\end{array}$