Calculate sin^-1(-√2/2) Step by Step

Step by step

The question

Question: find sin⁻¹(−√2/2).

sin⁻¹(−√22) = ?

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.

sin θ = −√22

2 The allowed range

Many angles have a sine of −√2/2.

Going round, there are many.

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.

−90° ≤ θ ≤ 90°

3 Find the angle

From the unit circle: sin 45° = √2/2.

Without the minus, the angle is 45°.

sin 45° = √22

Careful!

315° has the right sin, but it is not between −90° and 90°.

Below the x-axis on the right, inside the range, is −45°.

Careful: not θ = 315° ✗

Sin is the up value.

Minus means below the x-axis.

In the right half, that is turning 45° the other way: θ = −45°.

θ = −45°

4 In radians

In radians: −45 times π over 180.

That is −π/4.

θ = −45 × π180 = −π4

So sin⁻¹(−√2/2) = −45° = −π/4.

sin⁻¹(−√22) = −45° = −π4

5 Check

Let's check: sin −45° is −√2/2.

sin −45° = −√22

And −45° is inside the range.

So it is right!

−90° ≤ −45° ≤ 90°

Let's look back.

First: what it asks.

Then: the allowed range.

Then: find the angle.

Then: in radians.

Last: check.

Answer