# Trigonometric Equations – Formulas and Solutions

Equations which include the trigonometric functions of unknown angle are called **trigonometric equations**.

For example, sin x = 1/2, cos x = sin 2x, etc.

### Solution of a Trigonometric Equations

A solution of a solution of trigonometric equations to find the value of unknown angle which satisfies the equation. The solution of a **trigonometric equation(sin x = 0)** for which x = 0, π, 2π. This is called principal of trigonometric equation.

## Common Solutions for Trigonometric Equations

Trigonometrical equation |
Solutions |

sin x = 0 | x = nπ |

cos x = 0 | x = (nπ + π/2) |

tan x = 0 | x = nπ |

sin x = 1 | x = (2nπ + π/2) = (4n+1)π/2 |

cos x = 1 | x = 2nπ |

sin x = sin θ | x = nπ + (-1)nθ, where θ ∈ [-π/2, π/2] |

cos x = cos θ | x = 2nπ ± θ, where θ ∈ (0, π] |

tan x = tan θ | x = nπ + θ, where θ ∈ (-π/2 , π/2] |

sin2 x = sin2 θ | x = nπ ± θ |

cos2 x = cos2 θ | x = nπ ± θ |

tan2 x = tan2 θ | x = nπ ± θ |

## Trigonometric Equations Examples

Let us see some an example to have a better understanding of trigonometric equations, which is given below:

**Example 1: Find the general solution of sin 3x =0**

**Solution: **

⇒ Sin 3x = 0

⇒ 3x = nπ

⇒ x = nπ/3

**Example 2: sin 2x – sin 4x + sin 6x = 0**

**Solution:**Given: sin 2x – sin 4x + sin 6x = 0

⇒sin 2x + sin 6x – sin 4x = 0

⇒2sin 4x.cos 2x – sin 4x = 0

⇒sin 4x (2cos 2x – 1) = 0

⇒sin 4x = 0 or cos 2x = ½

⇒4x = nπ or 2x = 2nπ ± π/3

Therefore, the general solution for the given trigonometric equation is:

⇒x = nπ/4 or nπ ± π/6

**Example 3: Find the principal solution of the equation sin x = 1/2.**

Solution: Since we know that sin π/6 = 1/2

**∴** sin 5π/6

and sin (π – π/6) = sin π/6 = 1/2

Hence, the principal solutions are x =π/6 and x = 5π/6.