When you know the two legs of a right triangle, the inverse tangent gives you the angle in one step. This is the most common way arctan is used in school and on the job.
The method
- Identify the angle θ you want.
- Label the side opposite θ and the side adjacent to θ (the leg that touches θ but is not the hypotenuse).
- Divide: ratio = opposite ÷ adjacent.
- Take the inverse tangent: θ = arctan(ratio).
Example 1: sides 3 and 5
- ratio = 3 ÷ 5 = 0.6
- θ = arctan(0.6) ≈ 30.96° (0.5404 rad)
The other acute angle is 90° − 30.96° = 59.04°, and the hypotenuse is √(3² + 5²) = √34 ≈ 5.831.
Example 2: a wheelchair ramp
A ramp rises 0.5 m over a horizontal run of 6 m.
- ratio = 0.5 ÷ 6 ≈ 0.0833
- θ = arctan(0.0833) ≈ 4.76°
Many accessibility guidelines recommend a slope no steeper than 1:12, which is exactly this ratio, so the ramp is right on the limit.
Example 3: the height of a tree
You stand 20 m from a tree and the top appears 35° above the horizontal. Here you know the angle and want a side, so you use tangent rather than arctan: height = 20 × tan 35° ≈ 14.0 m. To go back from the sides to the angle, arctan(14.0 / 20) ≈ 35°.
Common mistakes
- Swapping the sides. arctan(5 ÷ 3) gives 59.04°, the other angle. Always put the side opposite your angle on top.
- Calculator in the wrong mode. If you expect degrees but get 0.5404, your calculator is in radians.
- Using the hypotenuse. Tangent only uses the two legs. If you know the hypotenuse and one leg, use arcsin or arccos instead.
Do it in one click
Choose Opp / Adj in the inverse tangent calculator, enter the two sides and you get the angle in nine units, the other angle, the hypotenuse and a labelled diagram.
Try it yourself: open the inverse tangent calculator to check any value from this article with full step-by-step working.