Inverse Trigonometric Functions
Domains, ranges, and key properties of arcsin, arccos, and arctan.
Learn inverse trig functions with worked examples and identities.
The Functions
y = arccos(x) means cos(y) = x
y = arctan(x) means tan(y) = x
Inverse trigonometric functions answer the question: "What angle has this sine (or cosine, or tangent) value?" Since trig functions are periodic and not one-to-one, each inverse function is restricted to a specific range to give a unique answer.
These functions are essential for finding angles in right triangles, converting between coordinate systems, and solving trig equations.
Domains and Ranges
| Function | Domain (Input) | Range (Output) |
|---|---|---|
| arcsin(x) | [−1, 1] | [−π/2, π/2] or [−90°, 90°] |
| arccos(x) | [−1, 1] | [0, π] or [0°, 180°] |
| arctan(x) | (−∞, +∞) | (−π/2, π/2) or (−90°, 90°) |
Key Identities
- arcsin(x) + arccos(x) = π/2 (for any x in [−1, 1])
- arctan(x) + arctan(1/x) = π/2 (for x > 0)
- arctan(−x) = −arctan(x)
- arcsin(−x) = −arcsin(x)
- arccos(−x) = π − arccos(x)
Derivatives
d/dx arccos(x) = −1 / √(1 − x²)
d/dx arctan(x) = 1 / (1 + x²)
Example 1
A ladder leans against a wall. The ladder is 5 m long and the base is 3 m from the wall. What angle does the ladder make with the ground?
The adjacent side is 3 m and the hypotenuse is 5 m
cos(θ) = adjacent / hypotenuse = 3/5 = 0.6
θ = arccos(0.6)
θ ≈ 53.13° (the ladder makes about a 53° angle with the ground)
Example 2
Verify the identity: arcsin(0.6) + arccos(0.6) = π/2.
arcsin(0.6) = 0.6435 radians (≈ 36.87°)
arccos(0.6) = 0.9273 radians (≈ 53.13°)
Sum = 0.6435 + 0.9273 = 1.5708 radians
π/2 = 1.5708 radians. The identity checks out: 36.87° + 53.13° = 90°.
Example 3
A ramp rises 2 m over a horizontal distance of 8 m. What is the angle of incline?
tan(θ) = opposite / adjacent = 2/8 = 0.25
θ = arctan(0.25)
θ ≈ 14.04° (a gentle incline suitable for wheelchair access, which requires ≤ 4.8°, so this ramp is too steep)
When to Use These
Inverse trig functions are used whenever you need to find an angle from a ratio.
- Solving right triangle problems in geometry and physics
- Navigation and surveying (finding bearings and elevation angles)
- Computer graphics (calculating rotation angles)
- Converting cartesian coordinates to polar coordinates
- Integration problems in calculus involving trig substitution