Cotangent Calculator

What cotangent is, in one ratio

Cotangent answers a plain question about a ramp or a roof: for each unit you climb, how far do you travel across? In a right triangle, cot(theta) = adjacent / opposite. Turn that upside down and you have tan(theta) = opposite / adjacent, so the two functions are reciprocals.

Written with sines and cosines, cot(theta) = cos(theta) / sin(theta). Both definitions are the same statement, because cos / sin = (adjacent / hypotenuse) divided by (opposite / hypotenuse), and the hypotenuse cancels out.

No unit survives that division. A 3-4-5 triangle has a cotangent of 4 / 3 = 1.3333 at the angle whose opposite side is 3, whether the sides are inches, feet or meters. That angle measures 36.87 degrees, since its tangent is 3 / 4 = 0.75 and the angle with a tangent of 0.75 is 36.87 degrees.

One check fits every answer: multiply the tangent by the cotangent and you should get 1. Here 0.75 x 1.3333 = 1.0000. The Texas A&M math team states the same ratio in its chapter on other trigonometric functions.

Two ways to compute it: 1 / tan and cos / sin

A calculator can reach cotangent two ways, and for most angles the two answers agree to the last digit. Route one divides 1 by the tangent: cot 30 = 1 / 0.5774 = 1.7321. Route two divides the cosine by the sine: 0.8660 / 0.5000 = 1.7321. Both land on the square root of 3.

The routes part company where a sine or cosine gets close to zero, because there a tiny rounding error gets multiplied by a huge number. At 90 degrees the cosine is 0 and the sine is 1, so cot 90 = 0 / 1 = 0 exactly. Push the same angle through the tangent instead and the answer depends on how the software rounds tan 90: some tools return a value near 1.6e16, so 1 / tan hands back 6.1e-17 instead of 0. The cotangent is still 0. The display is what went wrong.

At 0, 180 and 360 degrees the problem flips and no route gives a number, because sine is zero. Section 4.14 of the NIST Digital Library of Mathematical Functions defines cot as cos / sin and marks the zeros of sine as the poles of cotangent for exactly that reason.

So: use cos / sin when you start from an angle, and use 1 / tan when a tangent value is what you already have in hand.

No value at 0, 180 and 360 degrees

Cotangent repeats every 180 degrees, not every 360. That shorter period comes straight out of the ratio: adding 180 degrees flips the sign of both cosine and sine, so cos / sin comes back unchanged. cot 30 = 1.7321 and cot 210 = 1.7321, which is the same number one half turn later.

At every multiple of 180 degrees the sine is exactly zero, so cotangent has no value there. Those points are the vertical asymptotes on the graph. Walk toward 0 from above and the value climbs without a ceiling: cot 10 = 5.6713, cot 5 = 11.4301, cot 1 = 57.29, cot 0.1 = 572.96. Walk toward 180 from below and the same wall shows up with the opposite sign: cot 179 = -57.29 and cot 179.9 = -572.96.

Two sides of one point, opposite directions, no meeting place. That is why the answer reads "not defined" at 0, 180, 360 or any negative multiple of 180, instead of printing a large number.

The zeros work the other way around. Cotangent is 0 wherever the cosine is 0, at 90 and 270 degrees, because 0 / 1 = 0. So the zeros of cotangent sit where tangent has its asymptotes, and the asymptotes of cotangent sit where tangent crosses zero. Figure 1 shows the curve and both walls.

Cotangent and tangent, angle by angle

Reciprocals mean one table covers both functions. Read across any row and the two numbers multiply to 1, apart from the rows where one of them has no value at all.

Three patterns fall out of the table. The sign of cotangent always matches the sign of tangent: both are positive in the first and third quadrants, both negative in the second and fourth. The size runs opposite, so a large tangent means a small cotangent. And the blank rows are different rows: 90 and 270 have no tangent but a cotangent of 0, while 0, 180 and 360 have no cotangent but a tangent of 0.

Supplementary angles, the pair that adds to 180 degrees, have cotangents of equal size and opposite sign: cot 30 = 1.7321 while cot 150 = -1.7321. Complementary angles, the pair that adds to 90 degrees, swap roles: cot 30 = tan 60 = 1.7321. That swap is where the prefix in cotangent comes from.

Roof pitch: cotangent is the run per unit of rise

Carpenters describe a roof as rise over run, in inches per foot. A 4/12 roof climbs 4 inches for every 12 inches across, so its tangent is 4 / 12 = 0.3333 and its cotangent is 12 / 4 = 3.0. The cotangent is the number of inches of horizontal run that come with each inch of rise, which is the number a framing square is laid out around.

The angle at the eaves measures 18.43 degrees. Cotangent says nothing about the length of the rafter, since that needs the hypotenuse and belongs to cosecant. It only compares the two legs of the triangle.

Worked check: a shed roof rises 3 feet over a 10 foot run. The cotangent is 10 / 3 = 3.3333 and the pitch is 3 / 10 = 0.30, or 30 percent. The angle measures 16.70 degrees. Shorten the run to 8 feet with the same rise and the cotangent drops to 8 / 3 = 2.6667, while the angle rises to 20.56 degrees. Figure 2 draws the 4/12 case to scale.

Ramps, grades and the 1:12 limit

Ramp slope is usually written as a ratio, and a ratio of run to rise is a cotangent. A 1:12 ramp rises 1 inch for every 12 inches of run, so its cotangent is 12 and its grade is 1 / 12 = 0.0833, or 8.33 percent. The angle of that ramp measures 4.76 degrees.

The 2010 ADA Standards cap new ramp runs at that slope. Section 405.2 requires a running slope not steeper than 1:12, as the US Access Board guide to ramps and curb ramps explains. Steeper runs are limited to existing buildings and short rises.

Sizing example: a 30 inch rise at the maximum 1:12 slope needs 30 x 12 = 360 inches of run, which is 30 feet. The same rise at 1:8, the steepest slope the standards permit in an existing building, needs 30 x 8 = 240 inches, or 20 feet, and the angle goes from 4.76 to 7.13 degrees.

For any grade, divide the run by the rise. A driveway that climbs 2 feet over 24 feet has a cotangent of 12, the same as a legal ramp, and an 8.33 percent grade. A walkway that climbs 2 feet over 16 feet has a cotangent of 8 and a 12.5 percent grade, which is steeper than the ADA limit allows on a new ramp.

From sine or cosine to cotangent, without the angle

A sine plus the Pythagorean identity gives you the cotangent in two steps, with no inverse function needed. Take sin = 0.6. The cosine is the square root of 1 - 0.36 = 0.64, so cos = 0.8 on the positive side, and cot = 0.8 / 0.6 = 1.3333. That angle measures 36.87 degrees.

Sign needs care. A sine of 0.6 also belongs to 143.13 degrees, where the cosine is -0.8 and the cotangent is -1.3333. A ratio alone cannot pick the quadrant, so the calculator reports the base answer and the mirror angle beside it.

A cosine is shorter still: cot = cos / sin, and sin = the square root of 1 - cos squared. For cos = 0.28, sin = 0.96 and cot = 0.28 / 0.96 = 0.2917. That is a 7-24-25 right triangle scaled down by 25.

The identity cot squared + 1 = csc squared chains the three reciprocal functions together. Square 1.3333 and you get 1.7777, add 1 for 2.7777, and the square root of 2.7777 is 1.6667, which is 1 / 0.6, the cosecant of the same angle. The check passes.

Derivative, inverse and four mistakes

The derivative of cotangent is minus cosecant squared, a result listed with the other trig derivatives in section 4.16 of the NIST DLMF. At 45 degrees the cosecant is 1.4142, its square is 2, so the slope of the cotangent curve there is -2 radians per radian. That negative slope holds between every pair of asymptotes: cotangent always falls as the angle grows.

Mistake one is the unit. cot 1 in radians is 0.6421, while cot 1 degree is 57.29. The same four keystrokes give answers 89 times apart.

Mistake two is reading cot as the inverse. cot 0.5 radians is 1.8305. The angle whose cotangent is 0.5 measures 63.43 degrees, and the angle whose tangent is 0.5 measures 26.57 degrees. Those three numbers get confused constantly, and the -1 superscript in cot^-1 makes it worse, because it can mean the reciprocal or the inverse depending on the book.

Mistake three is expecting cot 90 to be undefined. It is 0, while tan 90 is the one with no value. Mistake four is a sign slip in the second or fourth quadrant, where sine and cosine disagree and the cotangent comes out negative even though the triangle looks ordinary. Near the asymptotes the sign is the only warning you get.