Tangent Calculator (tan and arctan)
Find tan θ for any angle in degrees, radians or multiples of π, or the angle from a tangent or slope, with exact values, cot θ and percent grade.
Get tan θ for any angle in degrees, radians or multiples of π, or enter a tangent (a slope) to find the angle with arctan. The calculator flags the angles where tangent is undefined, gives exact values for every multiple of 15°, and turns the result into a percentage grade, which is handy for ramps, roads and roof pitches as well as trigonometry homework.
How to use this tool
- Choose tan θ from an angle, or the angle from a tangent value (arctan).
- Enter the angle and its unit, or the tangent or slope value.
- Read tan θ, the exact form, cot θ and the grade.
- Use the table of common angles to check standard values.
Formula
tan θ = sin θ ÷ cos θ = opposite ÷ adjacent = rise ÷ run
Tangent repeats every 180°. It is positive in quadrants I and III, negative in II and IV, and undefined wherever cos θ = 0 (90°, 270° and so on). arctan returns the principal angle between −90° and 90°; add 180° for the other solution within a full turn. Grade in percent = tan θ × 100.
Example
A wheelchair ramp rises at 4.76°. tan 4.76° ≈ 0.083269, a grade of about 8.33%, which is the familiar 1:12 ramp ratio. Working backwards, a 1:12 slope has a tangent of 1 ÷ 12 ≈ 0.083333, and arctan gives an angle of about 4.763642°.
Frequently asked questions
Why is tan 90° undefined?
tan θ = sin θ ÷ cos θ, and cos 90° = 0, so the division has no value. On a graph the tangent shoots off towards infinity near 90°, a vertical asymptote. The same happens at 270°, −90° and every 180° from there.
How do I turn a roof pitch into an angle?
Divide the rise by the run and take arctan. A 6/12 pitch has a tangent of 0.5, so its angle is arctan(0.5) ≈ 26.565°.
Is tan⁻¹ the same as cot?
No. tan⁻¹ (arctan) is the inverse function and returns an angle. cot θ = 1 ÷ tan θ is the reciprocal ratio. Both appear here so you can see the difference.
Is a 100% grade vertical?
No. A 100% grade means the rise equals the run, which is 45°. A vertical line would need an infinite grade.
Last updated: 2026-10-30