# What is a Trigonometric Equation?

The equations having trigonometric functions of unknown angles are known as trigonometric equations.

For example: tan θ =

### Solution of a trigonometric equation

The value of the unknown angle that satisfies the given trigonometric equation is called the solution of the trigonometric equation.

## General Solutions of Trigonometric Equations

Under this section, we will learn about the general solutions of the trigonometric equations Sin θ = 0, Cos θ = 0, Tan θ = 0 and Cot θ = 0, Sin θ = Sin α, Cos θ = Cos α and Tan θ = Tan α and a Cos θ + b Sin θ = c

General Solution of Sin θ = 0

The general solution of Sin θ = 0 is θ = n Π, n ε Z

### General Solution of Cos θ = 0

The general solution of Cos θ = 0 is θ = (2n + 1) , n ε Z.

### General Solution of Tan θ = 0

The general solution of Tan θ = 0 is θ = n Π, n ε N.

General Solution of Cot θ = 0

The general solution of Cot θ = 0 is (2n + 1), n ε Z.**Important Note: **

As we have discussed the general solution of 4 trigonometric ratios out of 6.

### General Solution of Sec θ = 0

Since Sec θ ≥ 1, or Sec θ ≤ -1

Therefore, Sec θ = 0 does not have any solution.

### General Solution of Cosec θ = 0

Since Cosec θ ≥ 1, or Cosec θ ≤ -1

Therefore, Cosec θ = 0 does not have any solution.

### General Solution of Sin θ = Sin α

θ = n Π + (-1)^{n} n α, n ε Z

The equation Cosec θ = Cosec α is equivalent to Sin θ = Sin α. Thus, Cosec θ = Cosec α and Sin θ = Sin α have the same general solution.

### General Solution of Cos θ = Cos α

θ = 2 n Π ± α, where n ε Z.

The equation Sec θ = Sec α is equivalent to Cos θ = Cos α. Thus, Sec θ = Sec α and Cos θ = Cos α have the same general solution.

### General Solution of Tan θ = Tan α

θ = n Π + α, n ε Z

The equation Tan θ = Tan α is equivalent to Cot θ = Cot α. Thus, Tan θ = Tan α and Cot θ = Cot α have the same general solution.

### General Solution of Sin^{2} θ = Sin^{2} α

θ = n Π ± α, n ε Z.

### General Solution of Cos^{2} θ = Cos^{2} α

θ = n Π ± α, n ε Z.

### General Solution of Tan^{2} θ = Tan^{2} α

θ = n Π ± α, n ε Z.

### General Solution of a Cos θ + b Sin θ = c

a Cos θ + b Sin θ = c, where a, b, c ε R such that │c│ ≤ √a^{2} + b^{2}

In order to solve this type of equations, we reduce them in the form Cos θ = Cos α or Sin θ = Sin α.

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