Tangent Calculator
Enter an angle in degrees, radians, gradians, or milliradians and get the tangent instantly, along with all six related trigonometric values and the inverse-tan (arctan) lookup. The calculator shows each step of the arithmetic and flags the asymptote points such as 90° and 270° where the tangent is undefined.
Formula
Worked example
For theta = 45 deg: sin 45 deg = 0.707107 and cos 45 deg = 0.707107, so tan 45 deg = 1.000000. For theta = 60 deg: sin 60 deg = 0.866025, cos 60 deg = 0.500000, so tan 60 deg = sqrt(3) = 1.732051. Inverse example: arctan(1) = 45 deg (the principal angle between -90 deg and 90 deg).
What the tangent of an angle means
The tangent of an angle theta is defined as the ratio of its sine to its cosine: tan theta = sin theta divided by cos theta. In a right triangle this equals the length of the side opposite the angle divided by the side adjacent to it. On the unit circle, a point sits at coordinates (cos theta, sin theta), and the tangent is simply the y-coordinate divided by the x-coordinate, which also equals the slope of the line from the origin to that point. These two interpretations, the triangle ratio and the unit-circle slope, are equivalent and underpin every application from surveying to signal processing.
Four angle units: degrees, radians, gradians, and milliradians
Angles can be measured in four common units. Degrees split a full turn into 360 equal parts. Radians measure the arc on a unit circle: one full turn = 2 pi rad. Gradians (also called gon or grade) divide a full turn into 400 parts, so a right angle is exactly 100 grad. Milliradians are one-thousandth of a radian and are used in ballistics and surveying. To convert: multiply degrees by pi/180 to get radians; multiply degrees by 400/360 to get gradians; multiply radians by 1000 to get milliradians. Picking the wrong unit is the most frequent source of error, so confirm the selector matches your problem.
Where the tangent is undefined: vertical asymptotes
Tangent is undefined wherever cosine equals zero, because the formula divides by cos theta. That happens at 90 deg and 270 deg within one full turn, and more generally at every odd multiple of 90 deg (written 90 deg + 180k for integer k), which equals pi/2 + pi*k in radians or 100 + 200k in gradians. As the angle approaches one of these asymptote points the tangent grows without bound, racing toward positive infinity from one side and negative infinity from the other. The graph has a vertical break at each of these angles, and the calculator reports the result as Undefined rather than a misleadingly large number.
Inverse tangent (arctan): recovering an angle from its tangent
The inverse tangent function, written arctan or tan-1, takes a tangent value and returns the angle that produced it. Because tangent repeats every 180 degrees, each tangent value corresponds to infinitely many angles. The arctan function returns the unique principal angle between -90 deg and 90 deg (exclusive), and you add or subtract multiples of 180 deg to find every other angle with the same tangent. Switch the calculator to "tan(theta) to angle" mode, enter the tangent value, and choose the output unit you need. The principal angle, the radian equivalent, and the degree equivalent are all shown.
All six trigonometric values from one angle
Toggle "Show all 6 trig values" to compute the complete set alongside tan. Sine (sin) is the y-coordinate on the unit circle. Cosine (cos) is the x-coordinate. Cotangent (cot) is the reciprocal of tangent, equal to cos over sin. Secant (sec) is the reciprocal of cosine. Cosecant (csc) is the reciprocal of sine. The Pythagorean identities link them: sin squared + cos squared = 1, and 1 + tan squared = sec squared. Any one of the six values, together with the quadrant, completely determines the other five.
Periodicity, symmetry, and the CAST rule
Tangent has a period of 180 deg (pi radians), half the period of sine and cosine. Adding or subtracting 180 deg from an angle leaves its tangent unchanged. Tangent is also an odd function: tan(-theta) = -tan(theta). The CAST rule tells you which trig functions are positive in each quadrant: All in quadrant I, Sine only in quadrant II, Tangent only in quadrant III, Cosine only in quadrant IV. These facts let you reduce any angle to an equivalent principal value between -90 deg and 90 deg before computing.
Common tangent values, 0 to 360 degrees
| Angle (deg) | Angle (rad) | Angle (grad) | tan(theta) |
|---|---|---|---|
| 0 deg | 0 | 0 grad | 0 |
| 30 deg | pi/6 | 33.33 grad | 0.5774 (1/sqrt(3)) |
| 45 deg | pi/4 | 50 grad | 1 |
| 60 deg | pi/3 | 66.67 grad | 1.7321 (sqrt(3)) |
| 90 deg | pi/2 | 100 grad | Undefined |
| 120 deg | 2pi/3 | 133.33 grad | -1.7321 |
| 135 deg | 3pi/4 | 150 grad | -1 |
| 150 deg | 5pi/6 | 166.67 grad | -0.5774 |
| 180 deg | pi | 200 grad | 0 |
| 210 deg | 7pi/6 | 233.33 grad | 0.5774 |
| 225 deg | 5pi/4 | 250 grad | 1 |
| 240 deg | 4pi/3 | 266.67 grad | 1.7321 |
| 270 deg | 3pi/2 | 300 grad | Undefined |
| 315 deg | 7pi/4 | 350 grad | -1 |
| 360 deg | 2pi | 400 grad | 0 |
Exact and decimal tangents at key angles across a full turn. Gradians shown in the third column.
Frequently asked questions
Why is the tangent of 90 degrees undefined?
Tangent equals sine divided by cosine, and the cosine of 90 deg is exactly zero. Dividing by zero is undefined, so tan 90 deg has no finite value. As the angle approaches 90 deg the tangent grows without bound, toward positive infinity from below and negative infinity from above, which is why the graph has a vertical asymptote there. The same happens at 270 deg and every 180 deg beyond.
Should I enter my angle in degrees or radians?
Use whichever unit your problem is stated in and set the selector to match. Geometry and everyday problems are usually in degrees. Calculus, physics, and engineering often use radians. Surveying uses gradians or milliradians. If you have degrees but need radians, multiply by pi divided by 180. To go the other way, multiply radians by 180 divided by pi. Leaving the selector on the wrong unit is the most common mistake.
How do I find the angle if I already know the tangent?
Switch the calculator to "tan(theta) to angle" mode, enter the tangent value, and choose the output unit. The calculator applies arctan to return the principal angle between -90 deg and 90 deg. Because tangent repeats every 180 deg, add or subtract multiples of 180 deg (or pi, or 200 grad) to find every angle that shares the same tangent.
What is a gradians and when is it used?
A gradian (also called gon or grade) divides a full turn into 400 equal parts, so a right angle is exactly 100 grad and a straight line is exactly 200 grad. Gradians are used primarily in surveying and civil engineering because whole-number gradian angles correspond to convenient metric fractions of a right angle. Most modern calculators and programming languages support gradians alongside degrees and radians.
What are the six trigonometric functions and how are they related?
The six standard trig functions for an angle theta are: sin (opposite over hypotenuse on a unit circle, the y-coordinate), cos (adjacent over hypotenuse, the x-coordinate), tan (sin divided by cos, opposite over adjacent), cot (1/tan, cos divided by sin), sec (1/cos), and csc (1/sin). They satisfy: sin^2 + cos^2 = 1, 1 + tan^2 = sec^2, and 1 + cot^2 = csc^2. Toggle "Show all 6 trig values" to compute all of them at once.
How accurate is the tangent calculation?
The calculator uses the IEEE 754 double-precision floating-point arithmetic built into JavaScript, which is accurate to about 15 to 17 significant decimal digits. Results are displayed to 6 significant figures, which is more than sufficient for most engineering, physics, and academic uses. Near the asymptote at 90 deg (and its multiples) the calculator detects the near-zero cosine and reports Undefined rather than showing a very large inaccurate number.