# 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:

Solution:

Principal solution

*x*= 0 (cos 0 = 1)

âˆ´ General solution is

*x*= 2

*nÏ€*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:**