Inverse Cosine Calculator

What is an inverse cosine calculator?

The inverse cosine calculator answers the question “which angle has this cosine?”. The cosine function takes an angle and returns a ratio between -1 and 1. The inverse cosine, written as $\arccos$ or $\cos^{-1}$, reverses that operation: you give it a value $x$ in the interval $[-1, 1]$ and it returns the angle $\theta$ whose cosine equals $x$.

This calculator reports the result in two units at once: degrees and radians. That makes it handy whether you are working through a geometry problem in degrees or a calculus or physics problem in radians.

How does it work?

The cosine of an angle is the x-coordinate of the corresponding point on the unit circle. For every value of $x$ between -1 and 1 there are infinitely many angles with that cosine, so the arccosine is defined to return a single, principal value in the range:

$0 \le \theta \le 180^\circ \quad (0 \le \theta \le \pi \text{ radians})$

The relationship is:

$\theta = \arccos(x)$

Because cosine never leaves the interval $[-1, 1]$, any input outside that range has no corresponding real angle, and the calculator simply returns no result.

To convert the principal value from radians to degrees, multiply by $\frac{180}{\pi}$:

$\theta_{\deg} = \arccos(x) \times \frac{180}{\pi}$

Worked examples

• $\arccos(0.5) = 60^\circ$, which is about $1.0472$ radians ($\frac{\pi}{3}$).
• $\arccos(1) = 0^\circ$, or $0$ radians, since the cosine of a zero angle is 1.
• $\arccos(0) = 90^\circ$, or about $1.5708$ radians ($\frac{\pi}{2}$).
• $\arccos(-1) = 180^\circ$, or about $3.1416$ radians ($\pi$).

Entering a value such as $2$, which lies outside $[-1, 1]$, returns nothing because no real angle has a cosine greater than 1.

Practical notes

The arccosine appears whenever you need to recover an angle from a ratio. A common example is the dot-product formula for the angle between two vectors, where the cosine of the angle equals the dot product divided by the product of the magnitudes; taking the arccosine of that ratio gives the angle directly. It also shows up in the law of cosines when solving for an unknown angle of a triangle.

If you need the cosine of a known angle instead, work in the other direction with the trigonometry calculator. To switch a result between degrees, radians, and gradians, use the angle unit converter.