Sine takes infinitely many angles to the same value, so it has no inverse until its domain is restricted.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The three ranges
arcsin runs from −π/2 to π/2, arccos from 0 to π, and arctan from −π/2 to π/2.
A calculator gives one answer
The restricted range forces a single output. Other solutions exist and must be found separately.
Finding the rest, for sine
If sin θ = k is positive, the solutions are θ and π − θ, then plus any multiple of 2π.
For cosine
If cos θ = k, the solutions are θ and −θ, again plus multiples of 2π.
Read the interval
A question restricted to one turn wants two answers. One asking for all solutions wants a general form.
Domains restricted to make inverses exist
Sine takes each value infinitely often, so it must be restricted to [−π/2, π/2] to become invertible. Cosine is restricted to [0, π] and tangent to (−π/2, π/2). Those are conventions, chosen for symmetry and convenience.
The calculator returns one of many
sin⁻¹(0.5) gives π/6, but the equation sin x = 0.5 has infinitely many solutions. Recovering the rest requires the unit circle and the period, and the exam expects all solutions in the stated interval.
The notation is ambiguous
sin⁻¹x is the inverse function; (sin x)⁻¹ is the reciprocal. The superscript means different things in the two positions, which is why many texts prefer arcsin.
Compositions do not always cancel
sin(sin⁻¹x) = x on the valid domain, but sin⁻¹(sin x) returns x only when x lies in the principal range. Checking whether the input falls in that range before cancelling is essential.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, 0)
Step 3: Watch an Example
One step at a time.
Watch Ines Find Every Solution
Ines solves sin θ = 0.5 for θ between 0° and 360°.
- Step 1
The inverse sine gives 30°, which is the reference angle.
Step 4: Your Turn
Practice makes it stick.
The Second Solution
Problem 1 of 2
sin θ = 0.5 gives 30°. What is the other solution below 360°, in degrees?
degrees
The Range
Problem 2 of 2
How many degrees does the range of arccos span?
Back to the Angle
1 of 8
sin θ = 1. What is θ in degrees?
2 of 8
cos θ = 1. What is θ in degrees?
3 of 8
cos θ = −1. What is θ in degrees?
4 of 8
sin θ = 0.866 gives 60°. The other solution below 360°?
5 of 8
sin θ = 2. How many solutions?
6 of 8
sin θ = 0.5 with θ between 0° and 720°. How many solutions?
7 of 8
Match each inverse function with its range.
Tap a card on the left to start.
8 of 8
sin θ = 0.643 gives 40°. The other solution below 360°?
Step 5: Quick Check
Show what you know.
Question 1 of 2
sin θ = 0.5 gives 30°. What is the other solution below 360°, in degrees?
Question 2 of 2
Why does a calculator return only one solution?
What You Learned
- An inverse trigonometric function needs a restricted range to exist.
- A calculator returns one solution; the others must be constructed.
- For sine use θ and π − θ; for cosine use θ and −θ.