Enter an angle in degrees or radians and get its tangent instantly, along with the reciprocal cotangent (cot) — undefined cases at 90°, 270°, 0°, and 180° clearly labeled.
Enter an angle to see its tangent and cotangent
This tan calculator finds the tangent of any angle you enter, in degrees or radians, and shows the reciprocal ratio, cotangent (cot), right alongside it. Unlike sine and cosine, tangent is genuinely undefined at certain angles — 90°, 270°, and every angle coterminal with them — because tan θ = sin θ / cos θ and cosine equals zero there. This calculator checks for that condition explicitly and displays "Undefined" instead of "Infinity" or a distractingly huge number, so you always get an honest, accurate result.
Tangent is the ratio of the opposite side to the adjacent side in a right triangle, or equivalently sin θ divided by cos θ on the unit circle — geometrically, it's the slope of the line from the origin to the point at angle θ. This calculator converts your angle to radians internally, evaluates sin and cos, divides them (after checking cos isn't near zero), then computes cot(θ) = cos(θ)/sin(θ) with its own zero-denominator check.
This tool is built for trigonometry and precalculus students checking tangent homework, surveyors and civil engineers calculating height from an angle of elevation, road and civil engineers computing slope or grade percentages, architects working out roof pitch, and anyone who needs a quick, reliable tangent value without digging out a scientific calculator.
Tangent is the trigonometric ratio most directly tied to slope — "rise over run" for a line at a given angle is exactly tan θ. This makes it the go-to function whenever a real-world problem involves an incline, elevation angle, or grade, and it behaves very differently from sine and cosine: instead of staying bounded between -1 and 1, tangent grows without limit as the angle approaches its undefined points.
Civil engineers and road designers express a road's steepness as a percentage grade derived from tan θ. Surveyors use tangent with a known baseline distance and a measured angle of elevation to find a building or tree's height. Architects calculate roof pitch as a tangent ratio. Physicists use tangent to find the angle of a resultant vector from its components. Photographers and cinematographers use tangent-based field-of-view calculations.
How tangent is defined, and its reciprocal relationship with cotangent
Unlike sine and cosine, tangent can be any real number — it grows without limit near its undefined angles.
Tangent breaks down wherever cosine is zero, since tan θ = sin θ / cos θ requires dividing by cos θ.
Cotangent flips tangent upside down (cot = 1/tan), so it is undefined exactly where tangent equals zero.
From entering an angle to reading the tangent and cotangent
Type the angle value into the input field — any real number, positive or negative, is accepted.
Select which unit your angle is measured in using the toggle above the input.
The calculator converts your angle to radians internally and evaluates tangent as sine divided by cosine, checking for a zero denominator first.
See the tangent result, or "Undefined" if the angle is 90°, 270°, or coterminal with them.
Review the reciprocal cotangent value, which is labeled "Undefined" whenever the angle is 0°, 180°, 360°, or coterminal with them.
Finding the tangent and cotangent of a 45° angle
A ramp rises at a 45° angle. What is the tangent of that angle (its slope), and what is its reciprocal cotangent?
Undefined cases: at 90°, cos 90° ≈ 0, so tan 90° is Undefined (while cot 90° = 0 is perfectly valid). At 0°, sin 0° = 0, so cot 0° is Undefined (while tan 0° = 0 is perfectly valid). The calculator shows exactly this mix rather than crashing or displaying "Infinity."
Domain, range, and period of tangent and cotangent
| Function | Domain | Range | Period |
|---|---|---|---|
| tan θ | All angles except 90°, 270°, ... | All real numbers | 180° |
| cot θ | All angles except 0°, 180°, 360°, ... | All real numbers | 180° |
Reading the result: the tangent value tells you the slope of the ray at your angle — a tangent of 1 means a 45° incline (equal rise and run), while very large tangent values mean the angle is approaching a near-vertical 90°.
Typical ranges: unlike sine and cosine, tangent and cotangent have no upper or lower bound — they can be any real number, and they swing from negative infinity to positive infinity across each 180° period.
Manual verification: for common reference angles, tan 0° = 0, tan 30° ≈ 0.5774, tan 45° = 1, tan 60° ≈ 1.7321, and tan 90° is undefined — a quick way to check the calculator's output by hand.
Where finding a tangent value is genuinely useful
Check tangent values for right-triangle and unit-circle problems.
Convert an incline angle into a percentage grade using tan θ.
Find a building or tree's height from a baseline distance and elevation angle in surveying.
Express a roof's steepness as a tangent-based rise-over-run ratio.
Compute the visible frame width from a lens angle and distance.
Verify a wheelchair ramp's slope meets required tangent-based limits.
Find a bearing angle from north and east distance components.
Compute gear tooth or cam profile angles that rely on tangent ratios.
Determine force direction ratios in angled support members.
Compute view angles and aiming trajectories using tangent-based math.
Find the angle of a resultant vector from its horizontal and vertical components.
Get an instant tangent value for any angle without a textbook or table.
What this tan calculator does well, and where it doesn't apply
How tangent compares to sine and cosine at 0°, 30°, 45°, 60°, and 90°
| Angle | tan | sin | cos |
|---|---|---|---|
| 0° | 0 | 0 | 1 |
| 30° | 0.577350 | 0.5 | 0.866025 |
| 45° | 1 | 0.707107 | 0.707107 |
| 60° | 1.732051 | 0.866025 | 0.5 |
| 90° | Undefined | 1 | 0 |
Summary: This tan calculator gives you an instant, free way to find the tangent of any angle plus its reciprocal cotangent, with undefined cases clearly flagged. Pair it with the Sin Calculator and Cos Calculator for a fuller trigonometry toolkit.
Common questions about the tangent function
Trusted educational references to go deeper on the tangent function
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