Inverse Trigonometric Functions Calculator
Compute arcsine, arccosine, or arctangent of any value and return the result in degrees or radians. Essential for finding angles in triangles, physics, and engineering from known ratios.
About this calculator
Inverse trigonometric functions reverse the standard trig operations: arcsin(x) returns the angle whose sine is x, arccos(x) returns the angle whose cosine is x, and arctan(x) returns the angle whose tangent is x. In symbols: if sin(θ) = x, then θ = arcsin(x). The domains are restricted to ensure a unique output — arcsin and arccos require −1 ≤ x ≤ 1, while arctan accepts all real numbers. The output ranges are: arcsin ∈ [−90°, 90°], arccos ∈ [0°, 180°], and arctan ∈ (−90°, 90°). To convert radians to degrees, multiply by 180/π. These functions are indispensable for solving triangles, analyzing slopes, computing phase angles in circuits, and determining directions in navigation.
How to use
Suppose you want to find the angle whose cosine is 0.6. Select 'arccos' as the function type and enter 0.6 as the input value. Choose 'degrees' as the output unit. The calculator computes: arccos(0.6) = cos⁻¹(0.6) ≈ 53.13°. If you prefer radians, switch the output unit to get approximately 0.9273 rad. Set precision to 4 decimal places for results like 53.1301°. This angle would represent, for instance, the tilt of a ramp where the horizontal run is 0.6 times the slope length.
Frequently asked questions
What is the domain and range of arcsin, arccos, and arctan?
Arcsin accepts input values from −1 to 1 and returns angles between −90° and 90° (or −π/2 to π/2 rad). Arccos also accepts −1 to 1 but returns angles between 0° and 180° (0 to π rad). Arctan accepts any real number and returns angles strictly between −90° and 90°. These restricted ranges ensure each function is single-valued (a proper mathematical function). Inputs outside these domains — such as arcsin(1.5) — are undefined in real numbers.
Why does arccos(x) give a different angle than arcsin(x) for the same input value?
Because arcsin and arccos have different output ranges and ask different questions. Arcsin(0.5) asks 'which angle between −90° and 90° has a sine of 0.5?' giving 30°. Arccos(0.5) asks 'which angle between 0° and 180° has a cosine of 0.5?' giving 60°. These two are actually complementary: arcsin(x) + arccos(x) = 90° for all valid x. This identity reflects the fact that in a right triangle, the two non-right angles always sum to 90°.
How do I convert an arctan result from radians to degrees?
Multiply the radian result by 180/π ≈ 57.2958. For example, arctan(1) = π/4 rad; multiplying by 180/π gives exactly 45°. Conversely, to convert degrees to radians, multiply by π/180. Most calculators and programming languages return inverse trig results in radians by default, so this conversion is frequently needed in engineering and geometry applications. This calculator handles the conversion automatically when you select 'degrees' as the output unit.