Solving Trigonometric Equations

General Solution

Contents

General Solution

Trigonometric Equations

Quiz

Solving Trigonometric Equations

Here are more examples on finding the general solution of trigonometric equations, as well as some useful techniques.

Find the general solution to 5 cos x = 9 sin x -2 cosec x

Solution:

5 cos x = 9 sin x - 2 cosec x

5 cos x = 9 sin x - 2/ sin x

5 cos x sin x = 9 sin²x - 2

5 cos x sin x = 9 sin²x - 2 (sin²x + cos²x)

7 sin²x - 5 sinx cos x - 2 cos²x = 0

(7 sin x + 2 cos x)(sin x - cos x) = 0

tan x = -2/7 or tan x = 1

x = 180^on + 164.1^o x = 180^on + 45^o

Find the general solution of sin 2x = sin x.

Solution:

sin 2x = sin x

2x = np + (-1)ⁿx

x = np/2 + (-1)ⁿx/2

We have to consider two possible cases, when n is even and when n is odd.

when n is even, that is, when n = 2k,

x = 2kp/2 + x/2

x = 2kp

when n is odd, that is, when n = 2k + 1

x = (2k + 1)p/2 -x/2

x = (2k + 1)p/3

Find the general solution of sin 2x = 2^1/2 sin x.

Solution:

sin 2x = 2^1/2 sin x

2 sinx cos x = 2^1/2 sin x

sinx (2^1/2 cosx - 1) = 0

sin x = 0 cos x = 2^-1/2

x = np x = 2np + 45^o or 2np - 45^o

Find the general solution of cos 2x + cos 4x = cos 3x

Solution:

cos 2x + cos 4x = cos 3x

cos3x cosx = cos 3x factor formula

cos3x (cos x - 1) = 0

cos x = 1 cos 3x = 0

x = 2np 3x = 2np + 90^o or 2np - 90^o

x = 2np/3 + 30^o or 2np/3 - 30^o