# Inverse Trigonometric Function

See how to solve the inverse trigonometric function

(arcsine, arccosine, arctangent).

3 examples and their solutions.

## Arcsine

### Formula

x = arcsin y

→ sin x = y (-π2 ≤ x ≤ π2)

Arcsine is the inverse function of sine.→ sin x = y (-π2 ≤ x ≤ π2)

(-π/2 ≤ x ≤ π/2)

So, to solve arcsin y,

1. Set x = arcsin y.

2. Change it to sin x = y.

(-π/2 ≤ x ≤ π/2)

3. Solve the trigonometric equation.

### Example

arcsin √32

Solution x = arcsin √32 - [1]

→ sin x = √32 (-π2 ≤ x ≤ π2)

- [2] [3]

→ sin x = √32 (-π2 ≤ x ≤ π2)

[1]

Set x = arcsin √3/2.

The goal is to find x.

The goal is to find x.

[2]

sin x = √3/2

SOH: Sine, Opposite side (√3), Hypotenuse (2)

→ Draw the right triangle

on a coordinate plane.

SOH: Sine, Opposite side (√3), Hypotenuse (2)

→ Draw the right triangle

on a coordinate plane.

[2]

-π/2 ≤ x ≤ π/2

→ Draw the right triangle on quadrant I or IV.

→ Draw the right triangle on quadrant I or IV.

↓

x = π3

[4]

Close

## Arccosine

### Formula

x = arccos y

→ cos x = y (0 ≤ x ≤ π)

Arccosine is the inverse function of cosine.→ cos x = y (0 ≤ x ≤ π)

(0 ≤ x ≤ π)

So, to solve arccos y,

1. Set x = arccos y.

2. Change it to cos x = y.

(0 ≤ x ≤ π)

3. Solve the trigonometric equation.

### Example

arccos (-12)

Solution x = arccos (-12) - [1]

→ cos x = -12 = -12 (0 ≤ x ≤ π)

- [2] [3]

→ cos x = -12 = -12 (0 ≤ x ≤ π)

[1]

Set x = arccos (-1/2).

The goal is to find x.

The goal is to find x.

[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.

[3]

0 ≤ x ≤ π

→ Draw the right triangle on quadrant I or II.

→ Draw the right triangle on quadrant I or II.

↓

↓

x = 2π3

[4]

(reference angle) = π/3

→ x = π - π/3 = 2π/3

→ x = π - π/3 = 2π/3

Close

## Arctangent

### Formula

x = arctan y

→ tan x = y (-π2 ≤ x ≤ π2)

Arctangent is the inverse function of tangent.→ tan x = y (-π2 ≤ x ≤ π2)

(-π/2 ≤ x ≤ π/2)

So, to solve arctan y,

1. Set x = arctan y.

2. Change it to tan x = y.

(-π/2 ≤ x ≤ π/2)

3. Solve the trigonometric equation.

### Example

arctan (-1)

Solution x = arctan (-1) - [1]

→ tan x = -1 = -11 (-π2 ≤ x ≤ π2)

- [2] [3]

→ tan x = -1 = -11 (-π2 ≤ x ≤ π2)

[1]

Set x = arctan (-1).

The goal is to find x.

The goal is to find x.

[2]

tan x = (-1)/1

TOA: Tangent, Opposite side (-1), Adjacent side (1)

→ Draw the right triangle

on a coordinate plane.

TOA: Tangent, Opposite side (-1), Adjacent side (1)

→ Draw the right triangle

on a coordinate plane.

[3]

-π/2 ≤ x ≤ π/2

→ Draw the right triangle on quadrant I or IV.

(This is why tan x = (-1)/1, not 1/(-1).)

→ Draw the right triangle on quadrant I or IV.

(This is why tan x = (-1)/1, not 1/(-1).)

↓

x = -π4

[4]

Close