Tangent Calculator

What this calculator does

So, you have an angle and you want its tangent. This tool takes the angle, in degrees or radians, and returns the tangent rounded to three decimal places, or tells you when the tangent is not defined.

One box for the angle, a dropdown for degrees or radians, and a button. That is the whole interface.

How to use it

1. Type your angle.
2. Choose degrees or radians to match it.
3. Press Calculate.

Match the unit to your value. At a few special angles the tangent has no value at all, and there the tool returns "Not Defined" rather than an enormous or misleading number.

What tangent actually is

In a right triangle, the tangent of an angle is the opposite side divided by the adjacent side, the last piece of SOHCAHTOA: Tangent equals Opposite over Adjacent. Notice the hypotenuse does not appear, so tangent compares the two shorter sides directly. There is another way to see it that ties everything together: tangent is sine divided by cosine. Since sine is the height of the unit-circle point and cosine is its width, their ratio, height over width, is the tangent.

Tangent as the slope of the angle

Height over width is exactly what slope means, and that is the most useful way to picture tangent: it is the steepness of the line drawn from the origin out to the angle. A tangent of 1 is a line rising at 45 degrees, climbing one unit up for one unit across. A tangent of 2 is twice as steep. This is the same slope idea as rise over run for a line, which is why the tangent of a line's angle of inclination equals its gradient. Unlike sine and cosine, which are trapped between -1 and 1, tangent has no ceiling: as the angle approaches straight up, the slope grows without any limit.

Why it is undefined at 90 degrees

This is the case sine and cosine never run into. Tangent is sine over cosine, and at 90 degrees the cosine is 0. Dividing by zero has no meaning, so the tangent of 90 degrees is undefined, not merely very large. The same thing happens at 270 degrees, and at every odd multiple of 90, all the points where the line has gone fully vertical and its slope can no longer be written as a number. The tool checks for exactly these angles and returns "Not Defined", which is the honest answer, rather than the huge rounding artefact a raw calculation would produce.

A worked example

Take 45 degrees. Here the opposite and adjacent sides are equal, so their ratio is 1, and the tangent of 45 degrees is 1.000. At 0 degrees there is no height, so the tangent is 0. At 60 degrees the tangent is about 1.732, a steeper line. And at 90 degrees, the tool returns Not Defined, since the cosine there is zero.

The tangent came from shadows

Sine and cosine grew out of the chords of a circle, but tangent and its partner cotangent came by a completely different road: measuring shadows. The length of the shadow a stick casts depends on the angle of the sun, and that relationship is a tangent. Early tables of tangents were really shadow tables, used with sundials and for finding the heights of towers and mountains from the shadows they threw. So while sine started life inside a circle, tangent started life on the ground, as the link between an angle and a length of shade.

Questions people ask

What does the tangent of an angle tell me?

In a right triangle, the opposite side divided by the adjacent side. Equivalently, sine divided by cosine, which is the slope of the line at that angle.

How is tangent related to slope?

They are the same idea. The tangent of a line's angle of inclination is its slope, the rise over the run.

Why is the tangent of 90 degrees undefined?

Because tangent is sine over cosine, and the cosine of 90 degrees is zero. Dividing by zero has no value, so the tangent does not exist there.

Why can tangent be so large?

Because it has no upper limit. As the angle nears 90 degrees the line approaches vertical, and its slope grows without bound, unlike sine and cosine which stay within -1 and 1.

What is the reciprocal of tangent?

Cotangent, which is 1 divided by tangent. The cotangent calculator handles that function.

References

On tangent as a ratio, a slope, and a shadow. Tangent is opposite over adjacent, equal to sine over cosine, and it grew historically from the measurement of shadows rather than from circle chords.

1. Eric W. Weisstein, "Trigonometric Functions," from MathWorld, a Wolfram resource, on the tangent and the other circular functions.
2. "Trigonometric functions," MacTutor History of Mathematics, on the tangent and cotangent arising from shadow lengths, separately from the sine.