Arcsine: Value
How to find the given arcsine value: formula, 1 example and its solution.
Formula
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).
Example
Example
Solution
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.