Inverse Trigonometric Functions
Definitions and ranges for arcsin, arccos, and arctan.
Find angles from trig values with inverse trig functions and worked examples.
The Functions
arcsin(x) — also written sin⁻¹(x) — finds the angle whose sine is x
arccos(x) — also written cos⁻¹(x) — finds the angle whose cosine is x
arctan(x) — also written tan⁻¹(x) — finds the angle whose tangent is x
Inverse trig functions do the opposite of regular trig functions.
Instead of taking an angle and giving a ratio, they take a ratio and give an angle.
Domains and Ranges
| Function | Input (Domain) | Output (Range) |
|---|---|---|
| arcsin(x) | -1 ≤ x ≤ 1 | -90° to 90° (-π/2 to π/2) |
| arccos(x) | -1 ≤ x ≤ 1 | 0° to 180° (0 to π) |
| arctan(x) | All real numbers | -90° to 90° (-π/2 to π/2) |
Key Relationships
- If sin(θ) = x, then arcsin(x) = θ
- If cos(θ) = x, then arccos(x) = θ
- If tan(θ) = x, then arctan(x) = θ
- arcsin(x) + arccos(x) = 90° for any x in [-1, 1]
Example 1
Find the angle whose sine is 0.5
θ = arcsin(0.5)
We need the angle where sin(θ) = 0.5
θ = 30° (or π/6 radians)
Example 2
A ramp rises 3 meters over a horizontal distance of 8 meters. What is the angle of inclination?
The angle can be found using tangent: tan(θ) = opposite / adjacent = 3 / 8
θ = arctan(3/8) = arctan(0.375)
θ ≈ 20.56°
When to Use It
Use inverse trig functions when:
- You know a trig ratio and need to find the angle
- Finding angles from side lengths in a right triangle
- Calculating angles of elevation or depression
- Solving trig equations for unknown angles
Key Notes
- Principal value ranges: arcsin: [−90°, 90°] or [−π/2, π/2]; arccos: [0°, 180°] or [0, π]; arctan: (−90°, 90°) or (−π/2, π/2). These restricted ranges ensure a unique output — trig functions are periodic and not one-to-one without restriction.
- arctan vs atan2: arctan(y/x) only returns an angle in the range (−90°, 90°) — it cannot distinguish between the second and fourth quadrants (e.g., (−1,−1) and (1,1) both give arctan = 45°). The two-argument function atan2(y, x) returns the correct angle in all four quadrants and is standard in programming.
- Derivatives for calculus: d/dx arcsin(x) = 1/√(1−x²); d/dx arccos(x) = −1/√(1−x²); d/dx arctan(x) = 1/(1+x²). These appear in integration formulas: ∫dx/√(1−x²) = arcsin(x) + C; ∫dx/(1+x²) = arctan(x) + C.
- Common identities: arcsin(x) + arccos(x) = π/2 for all x in [−1,1]. arctan(x) + arctan(1/x) = π/2 for x > 0. These can simplify expressions involving multiple inverse trig functions.
- Applications: Inverse trig functions are used to find angles from known ratios in surveying, robotics inverse kinematics, physics (angle of refraction, projectile launch angle from range/speed), 3D graphics (Euler angle extraction from rotation matrices), and navigation (bearing from coordinate differences).