# Trigonometric Equations

Definition:- Equations involving trigonometric functions as variables are called trigonometric equations.

Recollect the property of sin x and cos x repeating (periodic) after 2Ï€ and tan x repeating after an interval of Ï€.
Hence trigonometric equations will have infinite solutions.

Eg: cos x = 0

Among these innumerable answers, the solutions for which 0 â‰¤ x â‰¤ 2 Ï€ are called principal solution.

Example:
Find the principal solutions of the equation
Principal solutions are
The expression involving integer
'n' which gives all the solution for the trigonometric equation is called general solution

Example:

General Solution (1):

â€‹Example:
Solve:

Solution:
Principal solutions are
Sin is negative in III & IV quadrants

Note:- You can use any one of the principal solutions

General Solution(2):

Solution:-

Example:
Solve:

Solution:

Principal solution x = 0  (cos 0 = 1)
âˆ´ General solution is
x = 2nÏ€ Â± 0

General Solution(3):
Let tan x = tan y, where y is the principal solution

Example:
Solve:

Solution:
Principal Solutions
[Recollect tan is positive in I & III quadrants]
âˆ´ General solution is

Note:- For solving trigonometric equations involving sec, cosec and cot functions, convert them to sin, cos or tan and then use the relevant formula.

Example 1:
Solve:

Solution:

When
General Solution:
When assume Î² to be the principal value, ie (can be obtained from trigonometric Tables. general solution:

Example 2:

Solution:

[Note Principal solution for ]

Example 3:
Solve:

Solution: