Calculate sin^-1(√2/2)

Step by step

The question

Question: find sin⁻¹(√2/2).

1 What it asks

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

We call that angle θ.

So sin θ = √2/2.

2 The allowed range

Many angles have a sine of √2/2.

135° and 405° 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 45° = √2/2.

Careful!

Other angles have the same sin, like 135°.

But only 45° is in the range.

So the button gives 45°.

45° is in the range.

So θ = 45°.

4 In radians

In radians: 45 times π over 180.

That is π/4.

So sin⁻¹(√2/2) = 45° = π/4.

5 Check

Let's check: sin 45° is √2/2.

And 45° 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.