Calculate sin^-1(1/2)

Step by step

The question

Question: find sin⁻¹(1/2).

1 What it asks

sin⁻¹ of 1/2 means: the angle whose sine is 1/2.

We call that angle θ.

So sin θ = 1/2.

2 The allowed range

Many angles have a sine of 1/2.

150° and 390° work too.

The sin⁻¹ button must give just one.

So sin⁻¹ only gives angles from −90° to 90°: the right half of the circle.

There, every sine from −1 to 1 happens just once.

3 Find the angle

From the unit circle: sin 30° = 1/2.

Careful!

Other angles have the same sin, like 150°.

But only 30° is in the range.

So the button gives 30°.

30° is in the range.

So θ = 30°.

4 In radians

In radians: 30 times π over 180.

That is π/6.

So sin⁻¹(1/2) = 30° = π/6.

5 Check

Let's check: sin 30° is 1/2.

And 30° is inside the range.

So it is right!

Let's look back.

First: what it asks.

Then: the allowed range.

Then: find the angle.

Then: in radians.

Last: check.