Calculate sin^-1(-1/2) Step by Step

Step by step

The question

Question: find sin⁻¹(−1/2).

sin⁻¹(−12) = ?

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.

sin θ = −12

2 The allowed range

Many angles have a sine of −1/2.

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 30° = 1/2.

Without the minus, the angle is 30°.

sin 30° = 12

Careful!

330° has the right sin, but it is not between −90° and 90°.

Below the x-axis on the right, inside the range, is −30°.

Careful: not θ = 330° ✗

Sin is the up value.

Minus means below the x-axis.

In the right half, that is turning 30° the other way: θ = −30°.

θ = −30°

4 In radians

In radians: −30 times π over 180.

That is −π/6.

θ = −30 × π180 = −π6

So sin⁻¹(−1/2) = −30° = −π/6.

sin⁻¹(−12) = −30° = −π6

5 Check

Let's check: sin −30° is −1/2.

sin −30° = −12

And −30° is inside the range.

So it is right!

−90° ≤ −30° ≤ 90°

Let's look back.

First: what it asks.

Then: the allowed range.

Then: find the angle.

Then: in radians.

Last: check.

Answer