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)
→ 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)
→ 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)
→ 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