Step by step
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°