Tangent calculator
Enter an angle for tan θ, exact for standard angles and flagged as undefined where cosine is zero. The inverse tangent (arctan) modes turn a tangent value, the two legs of a right triangle, or a slope written as a gradient or percentage grade back into an angle.
Type 30, 30°, pi/6, π/3, 1.2 rad or 36°52′. Anything with π is read as radians.
Any real number. Roots work: √3, -1/√3.
Examples: 0.5, 3/4, -2, 8%.
The point at θ on the unit circle is (cos θ, sin θ) = (0.5, 0.866025). The yellow segment on the line x = 1 has length tan θ.
Show the working, step by step
60° is in quadrant I, where all six functions are positive, so it is its own reference angle.
reference angle = 60°
Evaluate tan directly (a standard angle, so the exact value is known).
tan 60° = √3 ≈ 1.73205
What tangent means
tan θ = opposite ÷ adjacent = sin θ ÷ cos θ
In a right triangle, tangent is the side opposite the angle divided by the side next to it: rise over run. That makes it the natural function for slopes. A line making angle θ with the horizontal has gradient m = tan θ. On the unit circle, tan θ is the length of the segment cut off on the vertical line x = 1, the yellow segment in the drawing, which is where the name (Latin tangens, touching) comes from.
Tangent: a worked example
The calculator opens with θ = 60°.
- 60° is in quadrant I, where tangent is positive.
- sin 60° = √3/2 and cos 60° = 1/2, so tan 60° = (√3/2) ÷ (1/2) = √3 ≈ 1.73205.
- As a slope, a 60° line rises 1.73205 for every 1 across, a 173.2% grade.
Where tangent is undefined
Tangent divides by cos θ, which is zero at 90°, 270° and every 90° + k·180° (π/2 + kπ in radians). At those angles the calculator reports “undefined” rather than a huge number. Between them the tangent takes every real value once, which is why its period is 180° and why arctan accepts any input.
Tangent values to know
| θ | Radians | tan θ exact | Decimal |
|---|---|---|---|
| 0° | 0 | 0 | 0 |
| 15° | π/12 | 2 − √3 | 0.267949 |
| 30° | π/6 | √3/3 | 0.57735 |
| 45° | π/4 | 1 | 1 |
| 60° | π/3 | √3 | 1.73205 |
| 75° | 5π/12 | 2 + √3 | 3.73205 |
| 90° | π/2 | undefined | – |
Slopes, grades and pitches
Every slope measure is a tangent in disguise. Choose “Slope or grade → angle” and type a gradient (0.5, 3/4) or a grade with a percent sign. The default, an 8% road, is arctan 0.08 = 4.57392°. Some reference points:
| Slope | Gradient | Angle |
|---|---|---|
| 1:12 wheelchair ramp | 0.0833 | 4.76° |
| 25% grade | 0.25 | 14.04° |
| 6 in 12 roof pitch | 0.5 | 26.57° |
| 100% grade | 1 | 45° |
| 1000% grade | 10 | 84.29° |
Tangent also measures heights you cannot reach. Standing 50 m from a building and sighting its top at 32° above the horizontal, the height above eye level is 50 × tan 32° = 31.2 m.
Common mistakes
- Reading a grade as an angle. A 10% slope is 5.71°, not 10°; a 100% slope is 45°.
- Missing the second solution. tan θ = k also holds 180° after the principal value.
- Radian mode. tan 1 is 1.55741 in radians and 0.0174551 in degrees.
Common questions
Why is tan 90° undefined?
tan θ = sin θ ÷ cos θ, and cos 90° = 0, so the division has no answer. Close to 90° the tangent grows without limit: tan 89° = 57.29 and tan 89.9° = 572.96. The same happens at 270° and at every 90° + k·180°, which is where the graph of tan x has its vertical asymptotes.
How do I convert a slope or grade to an angle?
Take the inverse tangent of the gradient. A slope of 0.5 (rise 1 over run 2) is arctan 0.5 = 26.57°. For a percentage grade, divide by 100 first: an 8% road is arctan 0.08 = 4.57°. A 100% grade is 45°, not 90°.
What is the difference between tan⁻¹ and cot?
tan⁻¹ x means the inverse tangent, arctan x: the angle whose tangent is x. cot x is the reciprocal, 1 ÷ tan x. They are not the same: tan⁻¹(1) = 45°, while cot(1 rad) = 0.642.
How many solutions does tan θ = k have?
Two in every full turn, 180° apart, for any real k. tan θ = 1 gives θ = 45° and 225°, and the general solution is θ = 45° + 180°n. Tangent repeats every 180° (π radians), not every 360°.
What is a 4 in 12 roof pitch in degrees?
A rise of 4 over a run of 12 is a gradient of 4/12 = 0.3333, so the angle is arctan(0.3333) = 18.43°. A 6 in 12 pitch is 26.57°, and a 12 in 12 pitch is 45°.
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Including cot θ, the reciprocal of the tangent.