Solving Arcsine Functions

Solving Arcsine Functions

How to solve arcsine function problems: definition, examples, and their solutions.

Definition

y = arcsin x is the inverse function of y = sin x (-pi/2 <= x <= pi/2).

y = arcsin x is the inverse function of y = sin x.

Of course the graph of y = sin x
is not one-to-one:
it fails the horizontal line test.

So, to define the inverse function,
the domain of y = sin x is fixed like this:
-π/2 ≤ xπ/2.

Graphing sine functions

Example 1

Find the value of the given expression. sin (arcsin 0.4)

Sine and arcsine functions
are each other's inverse function.

So sin (arcsin 0.4) = 0.4.

Inverse functions

Example 2

Find the value of the given expression. arcsin 1/2

Set x = arcsin 1/2.

Next, sine both sides.
(left side) = sin x
(right side) = sin (arcsin 1/2)
= 1/2

So sin x = 1/2.
(-π/2 ≤ xπ/2)

To solve sin x = 1/2,
draw a right triangle that satisfies
sin x = 1 / 2.

Sine: SOH.
So sin x = (opposite side) / (hypotenuse) = 1 / 2.

The right triangle is a 30-60-90 triangle.

So (adjacent side) = √3.

So x = 30º (⋅π/180)
= π/6.

Radian measure