Inverse Trig Calculator (arcsin, arccos, arctan)

Understanding the inverse trigonometric functions calculator — concept and principle

Inverse trigonometric functions answer the reverse question: given a sine, cosine or tangent value, which angle produced it? This calculator returns arcsin, arccos and arctan of a number in degrees, plus arcsin and arctan in radians. People use it to find an angle in a right triangle from two sides, to get a slope angle from a gradient, to find the angle between vectors from a dot product, and in physics problems such as refraction and inclined planes.

How it is calculated

• arcsin x: the angle in [−90°, 90°] whose sine is x. Requires −1 ≤ x ≤ 1.
• arccos x: the angle in [0°, 180°] whose cosine is x. Requires −1 ≤ x ≤ 1.
• arctan x: the angle in (−90°, 90°) whose tangent is x. Any real x is allowed.
• Degrees = radians × 180 ÷ π.

The principle

sin, cos and tan are periodic, so many angles share the same value (sin 30° = sin 150° = 0.5). To build a true inverse function, each one is restricted to an interval where it is one-to-one and covers every possible value. These intervals are the principal values listed above. On them, sin and tan increase and cos decreases, so each value corresponds to exactly one angle. A useful consequence is arcsin x + arccos x = 90° for every x in [−1, 1].

Worked example

1. Default input: value x = 0.5.
2. arcsin 0.5: the angle in [−90°, 90°] with sine 0.5 is 30°, which is π/6 = 0.5235987756 rad.
3. arccos 0.5: the angle in [0°, 180°] with cosine 0.5 is 60°. Check: 30° + 60° = 90°.
4. arctan 0.5 = 0.463647609 rad × 180/π = 26.56505118°.

How to read the result

The calculator returns only the principal angle. Other solutions exist: for sin θ = 0.5 they are θ = 30° + 360°k and θ = 150° + 360°k; for cos θ = 0.5 they are θ = ±60° + 360°k; for tan θ = 0.5 they are θ = 26.565° + 180°k. In a triangle, use the context to decide, for example whether the angle can be obtuse. arctan of a slope expressed as rise ÷ run gives the angle of inclination: 0.5 means a 26.6° incline.

Common mistakes

• Entering a value outside [−1, 1] for arcsin or arccos, which has no real answer.
• Reading sin⁻¹ x as 1/sin x. sin⁻¹ x means arcsin x; the reciprocal is csc x.
• Forgetting the second solution when solving sin θ = k on 0°–360°.
• Using arctan(y/x) for a direction angle; it cannot tell opposite quadrants apart, so use atan2 instead.

Frequently asked questions

Why does arcsin(2) give no result?

The sine of a real angle is never larger than 1, so no real angle has sine 2.

How do I find an angle of a right triangle from two sides?

Use arctan(opposite ÷ adjacent), arcsin(opposite ÷ hypotenuse) or arccos(adjacent ÷ hypotenuse).

Why is arccos not shown in radians?

Multiply the degree value by π/180: 60° = π/3 ≈ 1.0471975512 rad.

Sources: NIST Digital Library of Mathematical Functions, §4.23 (inverse trigonometric functions, principal values) · Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 4