# Arcsine: Value

How to find the given arcsine value: formula, 1 example, and its solution.

## Formula

Arcsine is the inverse function of sine.

So, to solve arcsine,

set x = arcsin y,

write sin x = y,

and solve the sine equation.

x is in (-π/2 ≤ x ≤ π/2).

y = sin x is not one-to-one.

But if (-π/2 ≤ x ≤ π/2),

y = sin x is one-to-one.

So its inverse function can be defined.

This is why x is in (-π/2 ≤ x ≤ π/2).

## Examplearcsin √3/2

set x = arcsin √3/2.

Then sin x = √3/2.

x is in (-π/2 ≤ x ≤ π/2).

Draw a right triangle

that satisfies

sin x = √3/2 and (-π/2 ≤ x ≤ π/2).

-π/2 ≤ x ≤ π/2

So the right triangle should be in

either quadrant I or quadrant IV.

See sin x = √3/2.

Sine is SOH:

Sine,

Opposite side (√3),

Hypotenuse (2).

So draw a right triangle on a coordinate plane

whose opposite side is √3

and whose hypotenuse is 2.

Draw the angle x

that starts from the 3 o'clock position.

Find the missing side

by using the Pythagorean theorem:

[base]^{2} + (√3)^{2} = 2^{2}.

Then the base is 1.

This is a right triangle

whose sides are 1, √3, and 2.

So this is a 30-60-90 triangle.

So the central angle x is, 60º, π/3.

Radian Measure

So π/3 is the answer.