Steply

Inverse trig values

Trigonometry · 21 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. What it asks
  2. The allowed range
  3. Find the angle
  4. In radians
  5. Check

The steps for Calculate sin^-1(1/2). Every solution names its steps, so you always know what comes next.

Worked examples

Calculate sin^-1(1/2) Easy

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.

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.

−90° ≤ θ ≤ 90°

3 Find the angle

From the unit circle: sin 30° = 1/2.

sin 30° = 12

Careful!

Other angles have the same sin, like 150°.

But only 30° is in the range.

So the button gives 30°.

Careful: only the angle in the range

30° is in the range.

So 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
sin−1(12) = 30° = π6

▶ Watch it solved

Calculate sin^-1(-1/2) Medium

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
sin−1(−12) = −30° = −π6

▶ Watch it solved

Calculate arcsin(√3/2) in degrees Harder

The question

Question: find sin⁻¹(√3/2).

Give the angle in degrees.

sin−1(√32) = ?

1 What it asks

sin⁻¹ of √3/2 means: the angle whose sine is √3/2.

We call that angle θ.

So sin θ = √3/2.

sin θ = √32

2 The allowed range

Many angles have a sine of √3/2.

120° and 420° 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.

−90° ≤ θ ≤ 90°

3 Find the angle

From the unit circle: sin 60° = √3/2.

sin 60° = √32

Careful!

Other angles have the same sin, like 120°.

But only 60° is in the range.

So the button gives 60°.

Careful: only the angle in the range

60° is in the range.

So θ = 60°.

θ = 60°

So sin⁻¹(√3/2) = 60°.

sin−1(√32) = 60°

4 Check

Let's check: sin 60° is √3/2.

sin 60° = √32

And 60° is inside the range.

So it is right!

−90° ≤ 60° ≤ 90°

Let's look back.

First: what it asks.

Then: the allowed range.

Then: find the angle.

Last: check.

Answer
sin−1(√32) = 60°

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. sin^-1(1/2)
    Answer
    sin−1(12) = 30° = π6
  2. cos^-1(1/2)
    Answer
    cos−1(12) = 60° = π3
  3. tan^-1(1)
    Answer
    tan−1(1) = 45° = π4
  4. sin^-1(√2/2)
    Answer
    sin−1(√22) = 45° = π4
  5. cos^-1(√3/2)
    Answer
    cos−1(√32) = 30° = π6
  6. tan^-1(√3)
    Answer
    tan−1(√3) = 60° = π3
  7. sin^-1(1)
    Answer
    sin−1(1) = 90° = π2
  8. cos^-1(0)
    Answer
    cos−1(0) = 90° = π2

Medium

  1. sin^-1(-1/2)
    Answer
    sin−1(−12) = −30° = −π6
  2. cos^-1(-1/2)
    Answer
    cos−1(−12) = 120° = 2π3
  3. tan^-1(-1)
    Answer
    tan−1(−1) = −45° = −π4
  4. sin^-1(-√3/2)
    Answer
    sin−1(−√32) = −60° = −π3
  5. cos^-1(-√2/2)
    Answer
    cos−1(−√22) = 135° = 3π4
  6. tan^-1(-√3/3)
    Answer
    tan−1(−√33) = −30° = −π6
  7. cos^-1(-1)
    Answer
    cos−1(−1) = 180° = π

Harder

  1. arcsin(√3/2) in degrees
    Answer
    sin−1(√32) = 60°
  2. arccos(-√3/2)
    Answer
    cos−1(−√32) = 150° = 5π6
  3. tan^-1(√3/3) in degrees
    Answer
    tan−1(√33) = 30°
  4. sin^-1(-√2/2)
    Answer
    sin−1(−√22) = −45° = −π4
  5. sin^-1(2)
    Answer
    sin θ = 2 is impossible: no angle.
  6. cos^-1(-3)
    Answer
    cos θ = −3 is impossible: no angle.