Step by step
The question
Question: find sin⁻¹(1).
sin−1(1) = ?
1 What it asks
sin⁻¹ of 1 means: the angle whose sine is 1.
We call that angle θ.
So sin θ = 1.
sin θ = 1
2 The allowed range
Many angles have a sine of 1.
90° and 450° 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
Careful!
Other angles have the same sin, like 90°.
But only 90° is in the range.
So the button gives 90°.
Careful: only the angle in the range
90° is in the range.
So θ = 90°.
θ = 90°
4 In radians
In radians: 90 times π over 180.
That is π/2.
θ = 90 × π180 = π2
So sin⁻¹(1) = 90° = π/2.
sin−1(1) = 90° = π2
5 Check
Let's check: sin 90° is 1.
sin 90° = 1
And 90° is inside the range.
So it is right!
−90° ≤ 90° ≤ 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(1) = 90° = π2