# Trigonometric Equation

See how to solve a trigonometric equation

(sine/cosine/tangent equation).

3 examples and their solutions.

## Sine Equation

### Formula

sin x = k

→ x = nπ + (-1)

How to find:→ x = nπ + (-1)

^{n}⋅θ1. Find one root of the equation: x = θ.

2. Put this into x = nπ + (-1)

^{n}⋅θ.

### Example

sin x = 12

(0 ≤ x ≤ 2π)

Solution (0 ≤ x ≤ 2π)

sin x = 1/2

SOH: Sine, Opposite side (1), Hypotenuse (2)

→ Draw the right triangle

on a coordinate plane.

SOH: Sine, Opposite side (1), Hypotenuse (2)

→ Draw the right triangle

on a coordinate plane.

↓

Opposite side: 1, Hypotenuse: 2

→ Adjacent side: √3

→ 30-60-90 Triangle

→ Reference angle: 30° = π/6

Radian

→ Adjacent side: √3

→ 30-60-90 Triangle

→ Reference angle: 30° = π/6

Radian

↓

θ = π6 - [2]

x = nπ + (-1)

^{n}⋅π6 - [3]

x = 0⋅π + (-1)

^{0}⋅π6

= +π6( o ) - [4]

x = 1⋅π + (-1)

^{1}⋅π6

= π - π6

= 6π6 - π6

= 5π6( o ) - [5]

x = 2⋅π + (-1)

^{2}⋅π6

= 2π + π6

= 12π6 + π6

= 13π6( x ) - [6]

x = π6, 5π6

[1]

(reference angle) = π/6

→ (central angle) = π/6

→ (central angle) = π/6

[2]

π/6 is one of the roots.

→ Set θ = π/6.

→ Set θ = π/6.

[3]

Put n = 0, 1, 2, ... into the formula

to find the x

that are in 0 ≤ x ≤ 2π.

to find the x

that are in 0 ≤ x ≤ 2π.

[4]

x = π/6 is in 0 ≤ x ≤ 2π.

→ x = π/6

→ x = π/6

[5]

x = 5π/6 is in 0 ≤ x ≤ 2π.

→ x = 5π/6

→ x = 5π/6

[6]

x = 13π/6 is not in 0 ≤ x ≤ 2π.

→ x ≠ 13π/6

→ x ≠ 13π/6

Close

## Cosine Equation

### Formula

cos x = k

→ x = 2nπ ± θ

How to find:→ x = 2nπ ± θ

1. Find one root of the equation: x = θ.

2. Put this into x = 2nπ ± θ.

### Example

cos x = -√22

(0 ≤ x ≤ 2π)

Solution (0 ≤ x ≤ 2π)

cos x = -√22 = -1√2

cos x = -1/√2

CAH: Cosine, Adjacent side (-1), Hypotenuse (√2)

→ Draw the right triangle

on a coordinate plane.

CAH: Cosine, Adjacent side (-1), Hypotenuse (√2)

→ Draw the right triangle

on a coordinate plane.

↓

↓

θ = 3π4 - [2]

x = 2nπ ± 3π4 - [3]

x = 2⋅0⋅π ± 3π4

= ±3π4

→ x = 3π4 - [4]

x = 2⋅1⋅π ± 3π4

= 2π ± 3π4

= 8π4 ± 3π4

= 11π4, 5π4

→ x = 5π4 - [5]

x = 3π4, 5π4

[1]

(reference angle) = π/4

→ (central angle) = π - π/4 = 3π/4

→ (central angle) = π - π/4 = 3π/4

[2]

3π/4 is one of the roots.

→ Set θ = 3π/4.

→ Set θ = 3π/4.

[3]

Put n = 0, 1, 2, ... into the formula

to find the x

that are in 0 ≤ x ≤ 2π.

to find the x

that are in 0 ≤ x ≤ 2π.

[4]

x = 3π/4, -3π/4

Only 3π/4 is in 0 ≤ x ≤ 2π.

→ x = 3π/4

Only 3π/4 is in 0 ≤ x ≤ 2π.

→ x = 3π/4

[5]

x = 11π/4, 5π/4

Only 5π/4 is in 0 ≤ x ≤ 2π.

→ x = 5π/4

Only 5π/4 is in 0 ≤ x ≤ 2π.

→ x = 5π/4

Close

## Tangent Equation

### Formula

tan x = k

→ x = nπ + θ

How to find:→ x = nπ + θ

1. Find one root of the equation: x = θ.

2. Put this into x = nπ + θ.

### Example

tan x = √3

(0 ≤ x ≤ 2π)

Solution (0 ≤ x ≤ 2π)

tan x = √3 = √31

tan x = √3/1

TOA: Tangent, Opposite side (√3), Adjacent side (1)

→ Draw the right triangle

on a coordinate plane.

TOA: Tangent, Opposite side (√3), Adjacent side (1)

→ Draw the right triangle

on a coordinate plane.

↓

↓

θ = π3 - [2]

x = nπ + π3 - [3]

x = 0⋅π + π3

= π3( o )

x = 1⋅π + π3

= π + π3

= 3π3 + π3

= 4π3( o )

x = 2⋅π + π3

x = π3, 4π3

[1]

(reference angle) = π/3

→ (central angle) = π/3

→ (central angle) = π/3

[2]

π/3 is one of the roots.

→ Set θ = π/3.

→ Set θ = π/3.

[3]

Put n = 0, 1, 2, ... into the formula

to find the x

that are in 0 ≤ x ≤ 2π.

to find the x

that are in 0 ≤ x ≤ 2π.

Close