Inverse Tangent: Step-by-Step Guide
Arctangent, written arctan(x) or tan^-1(x), is the inverse of tangent. It answers 'what angle has this tangent?' We only allow answers strictly between -pi/2 and pi/2. You can put any real number in, and you get an angle between -pi/2 and pi/2 out.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Remember any number works. Unlike arcsin and arccos, arctan accepts any real number.\[x \in \mathbb{R}\]
- Recognize the tangent value. Compare x to common tangent values: 0, root 3 / 3, 1, root 3, and their negatives.\[\tan\theta \in \{0, \tfrac{\sqrt{3}}{3}, 1, \sqrt{3}\}\]
- Find the angle between -pi/2 and pi/2. Arctan is odd: arctan(-x) = -arctan(x). A positive x gives an angle between 0 and pi/2. A negative x gives an angle between -pi/2 and 0.\[\arctan(-x) = -\arctan(x)\]
- Check. Make sure tan(angle) = x and the angle is between -pi/2 and pi/2.\[\tan(\arctan(x)) = x\]
Worked example
Find arctan(-sqrt(3)).
Step 1: -root 3 is a valid real number.
Step 2: tan(pi/3) = root 3, so the reference angle is pi/3.
Step 3: The input is negative and arctan is odd, so the angle is -pi/3.
Step 4: Check: tan(-pi/3) = -root 3, and -pi/3 is between -pi/2 and pi/2. It works.
The answer is arctan(-root 3) = -pi/3.
Tips and common mistakes
- Arctan only gives values strictly between -pi/2 and pi/2. It never equals exactly -pi/2 or pi/2.
- Arctan is odd: arctan(-x) = -arctan(x).
- As x gets very large, arctan(x) gets close to pi/2. As x gets very negative, arctan(x) gets close to -pi/2.
- Common values: arctan(0) = 0, arctan(root 3 / 3) = pi/6, arctan(1) = pi/4, arctan(root 3) = pi/3.
- arctan(x) + arctan(1/x) = pi/2 when x is positive.
- In calculus, the derivative of arctan(x) is 1 / (1 + x^2).
- When you need the angle of a point in any quadrant, use atan2(y, x). Plain arctan can lose track of the quadrant.
- To find every solution of tan(theta) = k, use theta = arctan(k) + pi n.