Inverse Cosine: Step-by-Step Guide
Arccosine, written arccos(x) or cos^-1(x), is the inverse of cosine. It answers 'what angle has this cosine?' Because many angles have the same cosine, we only allow answers between 0 and pi. The input x must be between -1 and 1, and the output is an angle between 0 and pi.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Check the input is between -1 and 1. Outside of that range, arccos does not exist for real numbers.\[-1 \leq x \leq 1\]
- Recognize the value. Compare x to common cosine values: 1, root 3 / 2, root 2 / 2, 1/2, 0, and their negatives.\[\cos\theta \in \{1, \tfrac{\sqrt{3}}{2}, \tfrac{\sqrt{2}}{2}, \tfrac{1}{2}, 0\}\]
- Find the angle between 0 and pi. Negative inputs give angles between pi/2 and pi. Positive inputs give angles between 0 and pi/2. You can use arccos(-x) = pi - arccos(x).\[\arccos(-x) = \pi - \arccos(x)\]
- Check. Make sure cos(angle) = x and the angle is between 0 and pi.\[\cos(\arccos(x)) = x\]
Worked example
Find arccos(-1/2).
Step 1: -1/2 is between -1 and 1, so it is valid.
Step 2: cos(pi/3) = 1/2, so the reference angle is pi/3.
Step 3: We need an angle between 0 and pi with cosine -1/2. Cosine is negative in Quadrant II, so the angle is pi - pi/3 = 2 pi / 3.
Step 4: Check: cos(2 pi / 3) = -1/2, and 2 pi / 3 is between 0 and pi. It works.
The answer is arccos(-1/2) = 2 pi / 3.
Tips and common mistakes
- Arccos always gives an angle between 0 and pi. It is never negative.
- Arccos is not odd or even. Use arccos(-x) = pi - arccos(x) instead.
- Common values: arccos(1) = 0, arccos(root 3 / 2) = pi/6, arccos(root 2 / 2) = pi/4, arccos(1/2) = pi/3, arccos(0) = pi/2.
- A handy shortcut: arcsin(x) + arccos(x) = pi/2.
- cos(arccos(x)) = x always works. But arccos(cos(theta)) only equals theta if theta is between 0 and pi.
- In calculus, the derivative of arccos(x) is -1 / the square root of (1 - x^2).
- The input must be between -1 and 1. Anything outside has no answer.
- On the unit circle, arccos asks: 'What angle in the top half has this x-coordinate?'