Inverse Trig Calculator (Arcsin, Arccos, Arctan)

This inverse trig calculator finds \( \arcsin x \), \( \arccos x \) and \( \arctan x \). It returns the principal angle in radians, exactly when the answer is a standard angle (such as \( \frac{\pi}{6} \)) and as a decimal otherwise, plus the angle in degrees, the range the answer is taken from, and a check that applies the original function.

It’s useful for solving trig equations, finding angles in triangles, and checking the integrals and derivatives that produce inverse trig functions. It’s free and needs no sign-up.

How to use the inverse trig calculator

1. Enter x. Fractions such as -1/2, roots such as sqrt(3)/2, and decimals are accepted.
2. Pick arcsin, arccos or arctan.
3. Press Evaluate.
4. Read the angle in radians and degrees. If x is outside the allowed domain (for arcsin and arccos, \( -1 \le x \le 1 \)), the calculator says so.

Domains and ranges

Sine, cosine and tangent repeat forever, so each inverse picks one interval, the principal range, where the original function takes every value exactly once:

Function	Domain (x)	Range (angle)
\( \arcsin x \)	\( [-1, 1] \)	\( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)
\( \arccos x \)	\( [-1, 1] \)	\( [0, \pi] \)
\( \arctan x \)	all real numbers	\( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)

So \( y = \arcsin x \) means \( \sin y = x \) and \( y \) lies in \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \).

Worked example: arcsin(1/2)

Which angle in \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \) has sine \( \frac12 \)? From the 30-60-90 triangle, \( \sin\frac{\pi}{6} = \frac12 \), so

$$\arcsin\tfrac12 = \tfrac{\pi}{6} \approx 0.5236 \text{ rad} = 30^\circ$$

Note that \( \sin\frac{5\pi}{6} = \frac12 \) as well, but \( \frac{5\pi}{6} \) is outside the range, so it isn’t the arcsin.

Worked example: arccos of a negative number

\( \arccos\left(-\frac12\right) \): the range of arccos is \( [0, \pi] \), so a negative input gives an angle in Quadrant II. The reference angle is \( \frac{\pi}{3} \), so the answer is

$$\arccos\left(-\tfrac12\right) = \pi - \tfrac{\pi}{3} = \tfrac{2\pi}{3} = 120^\circ$$

Worked example: arctan(−1) and a non-standard value

\( \arctan(-1) = -\frac{\pi}{4} = -45^\circ \), because arctan returns angles between \( -90^\circ \) and \( 90^\circ \). For a value with no exact form, such as \( \arcsin 0.3 \), the calculator gives \( 0.30469 \) rad, or about \( 17.458^\circ \).

Tips and common mistakes

• \( \sin^{-1}x \) is not \( \frac{1}{\sin x} \). The \( -1 \) means inverse function. The reciprocal of sine is \( \csc x \).
• Only one answer. An equation like \( \sin\theta = \frac12 \) has infinitely many solutions; arcsin gives only the principal one. Add the others yourself: \( \theta = \frac{\pi}{6} + 2k\pi \) or \( \frac{5\pi}{6} + 2k\pi \).
• Out of domain. \( \arcsin 2 \) has no real value, since sine never exceeds 1.
• Undoing in the wrong order. \( \sin(\arcsin x) = x \) for every allowed x, but \( \arcsin(\sin x) = x \) only when x is already in the principal range.

Inverse trig in calculus

Inverse trig functions appear when you differentiate and integrate. The key results are

$$\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1 - x^2}}, \qquad \frac{d}{dx}\arctan x = \frac{1}{1 + x^2}$$

so \( \int \frac{dx}{1 + x^2} = \arctan x + C \). The full derivations are in the derivative of arcsin x and derivative of arctan x guides.

Related guides

• Inverse Trig Functions: Graphs, Derivatives and Integrals
• Derivative of arcsin(x)
• Derivative of arctan(x)
• The Unit Circle

Further reading

• Inverse trigonometric functions (Wikipedia) — notation, principal values, identities, derivatives and integrals.
• Inverse Trig Functions (Paul’s Algebra/Trig Review) — how to evaluate inverse trig functions and why the ranges are restricted.

FAQ

What is arcsin?

\( \arcsin x \) is the angle between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \) whose sine is x. It’s also written \( \sin^{-1}x \).

Why does arcsin only work between −1 and 1?

Sine only takes values in \( [-1, 1] \), so no angle has a sine of 2 or −3.

How do I get the answer in degrees?

The calculator shows both. To convert by hand, multiply radians by \( \frac{180}{\pi} \).

What is arctan of infinity?

As x grows without bound, \( \arctan x \) approaches \( \frac{\pi}{2} \) but never reaches it. Similarly it approaches \( -\frac{\pi}{2} \) as \( x \to -\infty \).