Inverse Trigonometric Functions Grapher
Visualize arcsin(x), arccos(x), and arctan(x) with customizable domains
Note: Inverse trig functions are restricted to their principal values to be functions.
Functions
Domain & Range
Appearance
Function Values at x = 0
Inverse Trigonometric Functions Explained
sin⁻¹(x) arcsin(x)
- Domain: [-1, 1]
- Range: [-π/2, π/2] radians ([-90°, 90°])
- Returns angle whose sine is x
- Undefined outside [-1, 1]
cos⁻¹(x) arccos(x)
- Domain: [-1, 1]
- Range: [0, π] radians ([0°, 180°])
- Returns angle whose cosine is x
- Undefined outside [-1, 1]
tan⁻¹(x) arctan(x)
- Domain: All real numbers
- Range: (-π/2, π/2) radians ((-90°, 90°))
- Returns angle whose tangent is x
- Horizontal asymptotes at y = ±π/2
Key Properties
Restricted Domains
Since trigonometric functions are periodic (repeat their values), we restrict their domains to create one-to-one functions that can be inverted. These restricted domains define the ranges of the inverse functions.
Derivatives
- d/dx [arcsin(x)] = 1/√(1-x²) for -1 < x < 1
- d/dx [arccos(x)] = -1/√(1-x²) for -1 < x < 1
- d/dx [arctan(x)] = 1/(1+x²) for all real x
Composition Identities
- sin(arcsin(x)) = x for x ∈ [-1, 1]
- arcsin(sin(x)) = x only for x ∈ [-π/2, π/2]
- Similar identities exist for cos and tan
Frequently Asked Questions
Why are inverse trig functions sometimes written with "arc" prefix?
The "arc" refers to the arc length on the unit circle that corresponds to the angle. Both notations (arcsin and sin⁻¹) are correct, but sin⁻¹ can sometimes be confused with (sin x)⁻¹ = 1/sin x, so many prefer "arcsin".
Can inverse trig functions return values outside their principal ranges?
No, by definition the inverse trig functions only return values in their principal ranges. For other values, you'd need to consider the periodic nature of trig functions and add multiples of π (180°).
How do I calculate inverse trig functions without a calculator?
For special angles (like 0, 1/2, √2/2, etc.), you can use known reference angles. For other values, you'd historically use tables or approximation methods. Today, calculators use sophisticated algorithms like CORDIC.
What's the difference between arctan(x) and arctan2(y,x)?
arctan(x) only returns angles between -π/2 and π/2 (-90° to 90°), while arctan2(y,x) considers the signs of both x and y to determine the correct quadrant, returning angles between -π and π (-180° to 180°).
Are inverse trig functions used in real-world applications?
Absolutely! They're used in physics (calculating angles from force components), engineering (control systems), computer graphics (rotation calculations), and navigation (converting between coordinate systems).