Domains of the inverse trigonometric functions
Each inverse function's domain is the range of the original function: [−1, 1] for sin⁻¹ and cos⁻¹, all of R for tan⁻¹ and cot⁻¹.
Domain comes from the range
An inverse function swaps inputs and outputs. So the domain of the inverse is the range of the original function. sin x and cos x only take values from −1 to 1, so only numbers in [−1, 1] can go into sin⁻¹ or cos⁻¹. tan x and cot x take every real value, so tan⁻¹ and cot⁻¹ accept every real number. sec x = 1/cos x and cosec x = 1/sin x are never strictly between −1 and 1, so sec⁻¹ and cosec⁻¹ accept only x ≤ −1 or x ≥ 1.
Domains at a glance
| Function | Domain | In words |
|---|---|---|
| sin⁻¹ x | [−1, 1] | from −1 to 1, ends included |
| cos⁻¹ x | [−1, 1] | from −1 to 1, ends included |
| tan⁻¹ x | R | every real number |
| cot⁻¹ x | R | every real number |
| sec⁻¹ x | R − (−1, 1) | x ≤ −1 or x ≥ 1 |
| cosec⁻¹ x | R − (−1, 1) | x ≤ −1 or x ≥ 1 |
How to find the domain of sin⁻¹(g(x)) or cos⁻¹(g(x))
- 1Write the condition: −1 ≤ g(x) ≤ 1.
- 2Solve both inequalities together, one step at a time.
- 3★If the expression has several parts, find each part's domain and keep only the common values (the intersection).
- 4Write the answer as an interval or a union of intervals.
★ key step
Example 1: a linear expression inside
Find the domain of sin⁻¹(2x − 1).
- Rule: sin⁻¹ u needs −1 ≤ u ≤ 1.
- −1 ≤ 2x − 1 ≤ 1
- Add 1 throughout: 0 ≤ 2x ≤ 2.
- Divide by 2: 0 ≤ x ≤ 1.
The domain is [0, 1].
Example 2: a square inside (board style)
Find the domain of cos⁻¹(x² − 2).
- Rule: cos⁻¹ u needs −1 ≤ u ≤ 1.
- −1 ≤ x² − 2 ≤ 1
- Add 2 throughout: 1 ≤ x² ≤ 3.
- x² ≥ 1 gives x ≤ −1 or x ≥ 1.
- x² ≤ 3 gives −√3 ≤ x ≤ √3.
- Keep the values that satisfy both conditions.
The domain is [−√3, −1] ∪ [1, √3].
Example 3: sec⁻¹ of an expression
Find the domain of sec⁻¹(2x + 1).
- Rule: sec⁻¹ u needs u ≤ −1 or u ≥ 1.
- 2x + 1 ≤ −1 gives 2x ≤ −2, so x ≤ −1.
- 2x + 1 ≥ 1 gives 2x ≥ 0, so x ≥ 0.
The domain is (−∞, −1] ∪ [0, ∞).
A calculator shows an error if you type sin⁻¹(1.2), because 1.2 is outside the domain [−1, 1]. Engineering software makes the same domain check before it works out the angle of a ramp or a roof.
1 marks: “Find the domain of sin⁻¹(2x).” −1 ≤ 2x ≤ 1, so −1/2 ≤ x ≤ 1/2. The domain is [−1/2, 1/2].
2 marks: “Find the domain of sin⁻¹(x² − 3).” −1 ≤ x² − 3 ≤ 1 gives 2 ≤ x² ≤ 4, so √2 ≤ |x| ≤ 2. The domain is [−2, −√2] ∪ [√2, 2].
2 marks: “Find the domain of f(x) = sin⁻¹ x + cos⁻¹(x − 1).” sin⁻¹ x needs −1 ≤ x ≤ 1. cos⁻¹(x − 1) needs −1 ≤ x − 1 ≤ 1, that is 0 ≤ x ≤ 2. The common values give the domain [0, 1].
Domain of the inverse = range of the original. sin and cos live in [−1, 1]; tan and cot reach every real number.