Arccosine Calculator

About Arccosine Calculator

Arccosine returns the angle whose cosine is x, in the range [0°, 180°].

How to use this calculator

1. Enter a cosine value between \(-1\) and \(1\).
2. Choose the angle unit your work uses, usually degrees or radians.
3. Read the inverse cosine result as the angle with that cosine value.
4. If needed, check the result by taking the cosine of the angle to see if you get back your original number.

The formula explained

The formula \(\arccos(x) = \cos^{-1}(x)\) computes the angle whose cosine is \(x\). For real-number inputs, \(x\) must be between \(-1\) and \(1\).

• x = the cosine value you want to invert, with \(-1 \le x \le 1\)
• \(\arccos(x)\) = the angle whose cosine equals \(x\)
• \(\cos^{-1}(x)\) = another notation for arccosine, not \(1/\cos(x)\)

Step by step method

1. Start with the cosine value you know, for example \(x = 0.5\).
2. Find the angle whose cosine equals that value, using a calculator or unit circle knowledge.
3. Check your answer by evaluating \(\cos(60^\circ) = 0.5\), so the arccosine of 0.5 is \(60^\circ\).

Worked example

Problem. Find \(\arccos(0.5)\) in degrees.

1. Set \(x = 0.5\).
2. Find the angle whose cosine is \(0.5\). Since \(\cos(60^\circ) = 0.5\), the angle is \(60^\circ\).
3. So \(\arccos(0.5) = 60^\circ\).

Tips and common mistakes

• Make sure the input is between \(-1\) and \(1\). Values outside that range do not have a real arccosine.
• Remember that \(\cos^{-1}(x)\) means arccosine, not the reciprocal \(1/\cos(x)\).

Frequently asked questions