Calculate tan of any angle in degrees or radians with exact common values.
Common tangent values
| Angle | Radians | Exact tan | Decimal |
|---|---|---|---|
| 0° | 0 | 0 | 0 |
| 30° | π/6 | 1/√3 = √3/3 | 0.57735027 |
| 45° | π/4 | 1 | 1.00000000 |
| 60° | π/3 | √3 | 1.73205081 |
| 90° | π/2 | — | Undefined |
| 120° | 2π/3 | −√3 | −1.73205081 |
| 135° | 3π/4 | −1 | −1.00000000 |
| 150° | 5π/6 | −1/√3 | −0.57735027 |
| 180° | π | 0 | 0 |
| 210° | 7π/6 | 1/√3 | 0.57735027 |
| 225° | 5π/4 | 1 | 1.00000000 |
| 240° | 4π/3 | √3 | 1.73205081 |
| 270° | 3π/2 | — | Undefined |
| 300° | 5π/3 | −√3 | −1.73205081 |
| 315° | 7π/4 | −1 | −1.00000000 |
| 330° | 11π/6 | −1/√3 | −0.57735027 |
| 360° | 2π | 0 | 0 |
Definition and geometry
What is the tangent of an angle?
In a right triangle, the tangent of an acute angle θ is the ratio of the side opposite the angle to the side adjacent to it: tan(θ) = opposite / adjacent. If a triangle has opposite = 3 and adjacent = 4, then tan(θ) = 3/4 = 0.75, which means θ ≈ 36.87°.
On the unit circle, tan(θ) equals the y-coordinate divided by the x-coordinate of the point where the terminal side of angle θ crosses the circle. Equivalently, tan(θ) = sin(θ) / cos(θ).
The tangent is undefined whenever cos(θ) = 0 — that is, at 90°, 270°, 450° and every angle of the form 90° + 180°·k. At these points the graph of tan has vertical asymptotes.
Period, sign, and range
Key properties of the tangent function
Period: 180° (π radians). tan(θ + 180°) = tan(θ) — values repeat every half-turn, not every full turn like sin and cos.
Sign by quadrant: Q1 (0°–90°) positive, Q2 (90°–180°) negative, Q3 (180°–270°) positive, Q4 (270°–360°) negative.
Range: all real numbers. Unlike sin and cos, tangent is unbounded — it grows without limit as the angle approaches 90°.
Odd function: tan(−θ) = −tan(θ). Example: tan(−45°) = −1.
Frequently asked questions
What is tan of 30, 45, and 60 degrees?
These are the three most-used exact values. tan(30°) = 1/√3 = √3/3 ≈ 0.57735027. tan(45°) = 1 exactly, because in a 45-45-90 triangle the two legs are equal. tan(60°) = √3 ≈ 1.73205081. Each value doubles in a meaningful way: 30° and 60° come from a 30-60-90 triangle where sides are in the ratio 1 : √3 : 2.
Why is tan(90°) undefined?
Because tan(θ) = sin(θ) / cos(θ), and cos(90°) = 0 — division by zero is not defined in the real numbers. As θ approaches 90° from below, cos(θ) shrinks toward 0 while sin(θ) stays near 1, so the ratio grows without bound (→ +∞). Approaching 90° from above, it goes to −∞. The graph of tangent has a vertical asymptote at every 90° + 180°·k. This calculator returns "Undefined" for such inputs.
How do I convert between degrees and radians?
One full turn equals 360° or 2π radians, so 180° = π rad. To convert: radians = degrees × π / 180, and degrees = radians × 180 / π. Quick anchors: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π. One radian ≈ 57.29578°. Most programming languages (JavaScript, Python, C) use radians in their Math.tan function, so degree inputs must be converted first.
How does a calculator actually compute tan of an arbitrary angle?
Modern processors use the CORDIC algorithm or a Taylor/Padé series for sin and cos, then divide. The series tan(x) ≈ x + x³/3 + 2x⁵/15 + 17x⁷/315 + … converges for small angles; large angles are first reduced modulo π. For angles within about 10⁻¹⁰ radians of an asymptote (90°, 270°, …), this calculator reports "Undefined" because the true value is effectively infinite and any finite answer would mislead.
When do I use tangent in real life?
Tangent links angles to slope. If a road rises 1 m over 10 m of horizontal distance, its angle is arctan(1/10) ≈ 5.71°. Surveyors measure the tangent of the elevation angle to find heights: a tree 20 m away seen at 35° above the horizon is 20 × tan(35°) ≈ 14 m tall. Engineers use tangent for camera field of view, ramp grades, roof pitches, and projectile trajectories. In graphics and robotics, the related atan2(y, x) function converts Cartesian coordinates to polar angles.
What is the difference between tan and arctan?
Tangent takes an angle and returns a ratio; arctangent (arctan or tan⁻¹) takes a ratio and returns an angle. Example: tan(45°) = 1, so arctan(1) = 45°. Because tangent repeats every 180°, arctan is restricted to the range (−90°, 90°) to keep it a proper function. To find an angle in a different quadrant, add 180° as needed, or use the two-argument atan2(y, x) which handles the sign of x and y automatically.
This tool evaluates the tangent of any angle, accepting input in degrees or radians and returning the result to eight decimal places. Toggle between units with a single click, or pick a common angle (0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°) from the preset row to see its exact value alongside the decimal. Asymptotes at 90° and 270° are detected automatically and labelled Undefined with a short explanation, so division-by-zero never produces a misleading number. Two quick examples: tan(45°) = 1 exactly, because sin and cos are equal there; tan(60°) ≈ 1.73205081, which is √3 — useful for 30-60-90 triangle problems, roof pitches, and ramp grades. A reference table lists tangent values from 0° to 360°, and the computation panel shows how sin / cos produces the final number.