Summary TRIGONOMETRY EXAMPLES

Gr 12 TRIG EQUATIONS: Illustrative Examples
1 1
2sin x –1 = 0 sin x =
2
etc.

2 1 1
(2sin x – 1)(3sin x + 1) = 0 sin x = 2 or sin x = –
3
etc.

3 8sin2x – 2sin x – 1 = 0 (4sin x + 1)(2sin x – 1) = 0
1 1
so sin x =  or sin x = etc.
4 2

4 cos2x – 2sin x = 4 1 – sin2x – 2 sin x = 4
sin2x – 2 sin x – 3 = 0
(sin x – 3(sin x + 1) = 0 so sin x = 3 (reject) or sin x = –1 etc.
5
7cos2x – 3sin x.cos x = 0 cos x.(7cos x – 3 sin x) = 0
cos x = 0 or tan x = 7/3 etc.

6 2sin2x – sin x.cos x – cos2x = 0 (2sin x + cos x)(sin x – cos x) = 0
1
tan x = – or tan x = 1 etc.
2

7 3sin2x – sin 2x = cos2x 3sin2x – 2sin x cos x – cos2x = 0
(3sin x + cos x)(sin x – cos x) = 0
1
tan x = – or tan x = 1 etc.
3

8 cos 2x = 3cos x 2cos2x – 1 = 3cos x
2cos2x – 3cos x – 1= 0

3  17 3  17
cos x = (reject) or cos x = etc.
4 4

9 cos 2x = 1 + sin 2x 1 – 2sin2x = 1 + 2sin x.cos x
2sin2x + 2sin x. cos x = 0
2sin x (sin x + cos x) = 0 so sin x = 0 or tan x = –1 etc.

10 3 3
sin x.cos12º – cos x.sin12º = 2
sin(x – 12º) =
2
etc.

11 3cos 2x – sin 2x – 1 = 0 3(cos2x – sin2x) – 2sin x.cos x – (cos2x + sin2x) = 0
3cos2x – 3sin2x – 2sin x.cos x – cos2x – sin2x = 0
2 2
2cos x – 2sin x.cos x – 4sin x = 0 so 2(cos x – 2 sin x)(cos x + sin x) = 0
1
tan x = or tan x = – 1 etc.
2

12 sin 2x + cos x + 3 cos 2x = 3 + 3 sin x 2sc + c + 3(1 – 2s2) = 3 + 3s
2sc + c + 3 – 6s2 = 3 + 3s
2sc + c – 6s2 – 3s = 0
(2 sin x + 1)(cos x – 3 sin x) = 0
1 1
sin x =  or tan x = etc.
2 3