Arcsine: Value

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).

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 = 2².

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.