Inverse Trigonometric Functions Calculator
Calculate values of inverse trigonometric functions (arcsin, arccos, arctan) with step-by-step explanations. Understand the domain and range restrictions for each inverse trig function.
Function Domains
- arcsin(x): Domain [-1, 1], Range [-π/2, π/2]
- arccos(x): Domain [-1, 1], Range [0, π]
- arctan(x): Domain (-∞, ∞), Range (-π/2, π/2)
Calculation Results
| Function | arcsin(0.5) |
|---|---|
| Result | 0.5236 radians (30°) |
| Exact Value | π/6 |
Step-by-Step Solution
1. We know that sin(π/6) = 0.5
2. Therefore, arcsin(0.5) = π/6
3. π/6 radians is equivalent to 30 degrees
Inverse Trigonometric Functions
Inverse trigonometric functions are the inverse functions of the trigonometric functions with appropriately restricted domains. They are used to obtain an angle from any of the angle's trigonometric ratios.
Definitions
The principal inverses are defined as follows:
- arcsin (inverse sine): For -1 ≤ x ≤ 1, arcsin(x) is the angle θ whose sine is x, with -π/2 ≤ θ ≤ π/2.
- arccos (inverse cosine): For -1 ≤ x ≤ 1, arccos(x) is the angle θ whose cosine is x, with 0 ≤ θ ≤ π.
- arctan (inverse tangent): For any real x, arctan(x) is the angle θ whose tangent is x, with -π/2 < θ < π/2.
Graphs of Inverse Trigonometric Functions
The graphs of the inverse trigonometric functions are reflections of their corresponding trigonometric functions about the line y = x, with restricted domains:
Find arcsin(0.5)
Solution: We know that sin(π/6) = 0.5, so arcsin(0.5) = π/6 ≈ 0.5236 radians or 30°.
Properties of Inverse Trigonometric Functions
| Function | Domain | Range | Principal Value |
|---|---|---|---|
| arcsin(x) | [-1, 1] | [-π/2, π/2] | y = arcsin(x) ⇔ x = sin(y) |
| arccos(x) | [-1, 1] | [0, π] | y = arccos(x) ⇔ x = cos(y) |
| arctan(x) | (-∞, ∞) | (-π/2, π/2) | y = arctan(x) ⇔ x = tan(y) |
Common Values
| x | arcsin(x) | arccos(x) | arctan(x) |
|---|---|---|---|
| 0 | 0 | π/2 | 0 |
| 1/2 | π/6 | π/3 | π/6 |
| √2/2 | π/4 | π/4 | π/4 |
| √3/2 | π/3 | π/6 | π/3 |
| 1 | π/2 | 0 | π/4 |
Applications
Inverse trigonometric functions are used in many areas including:
- Geometry (finding angles in right triangles)
- Physics (wave functions, projectile motion)
- Engineering (signal processing, control systems)
- Computer graphics (rotation calculations)
- Navigation (bearing calculations)