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Cotangent Calculator

Range: -1,000,000 ° – 1,000,000 °

Result

1.0000

Cotangent

A cotangent calculator returns the cotangent of an angle — the reciprocal of the tangent, which is also the ratio of the side next to the angle to the side across from it. Of the six ratios a right triangle offers, this is the one that flips the tangent over: where the tangent is opposite over adjacent, the cotangent is adjacent over opposite, so the two are one divided by the other at every angle where both exist. That reciprocal relationship is worth holding on to, because it explains everything odd about this function in one step. The tangent runs out at a right angle, where the adjacent side has shrunk to nothing; the cotangent runs out at zero degrees, where it is the opposite side that shrinks. Move the angle to a right angle and the roles swap — the tangent has no value and the cotangent is zero; move it to zero and the cotangent has no value while the tangent is zero. This page therefore refuses exactly the angles the tangent page accepts, and answers exactly the angles it refuses. The table below prints both columns so you can see that happen rather than take it on trust. Two other things follow from the definition. The cotangent is an odd function, so a negative angle gives the negative of the answer; and it repeats every half turn rather than every full one, like the tangent, because flipping both the adjacent and the opposite sides over leaves their ratio alone. Forty-five degrees is the one angle where the reading is a whole number — exactly one, since the two legs of that triangle are equal — and everything else is a reciprocal of a root.

The tangent and the cotangent side by side at twelve angles

θ (degrees)tan θcot θ
00—
300.57741.7321
4511
601.73210.5774
90—0
120-1.7321-0.5774
135-1-1
150-0.5774-1.7321
1800—
2100.57741.7321
270—0
330-0.5774-1.7321

Two columns and twelve angles, and the whole reason for printing both is the two dashes: they sit in the tangent column at ninety and two hundred and seventy degrees, and in the cotangent column at zero and one hundred and eighty. Nowhere do the two functions fail together, and that is the reciprocal relationship stated as a picture. Read the rows in pairs and the rest of the pattern appears too. At forty-five degrees both columns hold one, which is what it means for a number to be its own reciprocal. At thirty and at sixty the two columns hold each other's answers, because the cotangent of an angle equals the tangent of its complement. And wherever one column is large the other is small — at thirty degrees, 0.5774 against 1.7321 — since the two multiply to one.

Formula

cot θ = 1 ÷ tan θ cot θ = adjacent ÷ opposite cot θ = cos θ ÷ sin θ

θ
The angle, in degrees or radians. It can be negative and it can be larger than a full turn — only where it lands on the circle matters
adjacent / opposite
The two legs of a right triangle with the angle θ in it: the side next to θ and the side across from it. The cotangent is the first over the second, which is the tangent upside down
tan θ
The tangent of the same angle. Dividing one by it gives the cotangent wherever it is not zero — at a right angle the tangent has no value, and the cotangent there is zero instead

Use this page when the ratio you want is the one that puts the adjacent side on top — a slope expressed as run over rise rather than rise over run, a rate of climb quoted the other way round, an angle you know in degrees and want as the reciprocal of its tangent. The cotangent table below gives both columns, so it is also the quickest way to see how the tangent and the cotangent trade places: one column is a dash exactly where the other holds a zero. If you want the tangent rather than its reciprocal, and the sine and cosine as well, the trigonometry page prints all three together and the numbers agree to the last decimal. And if what you have is a cotangent and you want the angle back, that is the inverse direction, which is a different question with more than one answer.

Worked examples

  1. An angle of 45 degrees, the one whole-number answer

    1. At 45 degrees the two legs of the right triangle are equal
    2. The cotangent is the adjacent side over the opposite side
    3. Equal quantities divided give exactly one, so the reading is 1 and not 0.9999

    The angle the page loads with, and the only angle at which this function produces an integer. Forty-five degrees is also where the tangent is one, which is no accident: if a number is its own reciprocal, the function and its reciprocal agree there. Every other reading on this page is either a root or the reciprocal of one, which is why the table below is written to four decimal places.

  2. An angle of 30 degrees

    1. The tangent of 30 degrees is √3/3, about 0.5774
    2. The cotangent is one divided by that, which is √3
    3. Rounded to four decimal places, √3 is 1.7321

    The cotangent of thirty degrees is the same number as the tangent of sixty degrees, and that is a rule rather than a coincidence: the two acute angles of a right triangle add to ninety, and the side across from one is the side next to the other. So the cotangent of an angle is always the tangent of its complement, and the table below shows the pair on adjacent rows.

  3. An angle in the second quadrant, 135 degrees

    1. One hundred and thirty-five degrees is 45 degrees past a right angle
    2. The point is in the second quadrant, so the adjacent side is now negative and the opposite side is still positive
    3. A negative divided by a positive is negative, so the cotangent is −1

    Exactly the negative of the forty-five degree reading, and that is the pattern for the whole second quadrant: the size of the cotangent is unchanged from the first quadrant angle folded onto it, and only the sign moves. The absolute value returns to one here because the triangle is the same shape as at forty-five degrees, merely placed differently.

  4. An angle of 90 degrees, where the cotangent is zero

    1. At a right angle the adjacent side has shrunk to zero
    2. The cotangent is the adjacent side over the opposite side, so it is zero over a positive number
    3. Zero divided by anything non-zero is zero, so the reading is 0

    This is the angle the tangent page refuses, and here it is an ordinary reading of zero. Nothing is being special-cased: the ratio has a zero on top rather than underneath, and dividing zero by a number is allowed. It is the single clearest illustration of what the reciprocal relationship does — the two functions do not merely differ in size at this angle, they trade the defined and the undefined.

Limitations

Three limits worth stating plainly. The readings are given to four decimal places and most of them are irrational: the cotangent of thirty degrees is √3, so 1.7321 is a truncation rather than the answer. Forty-five degrees is the only angle on this page with an exact, terminating value, which is why the table looks uneven — some rows land on clean numbers and the rest do not. Second, the cotangent has no value at zero degrees, at one hundred and eighty degrees, and at every angle that reduces to either, including negative angles and angles past a full turn; this page refuses all of them, and the same refusal covers any angle you reach by adding a whole turn to those two. Third, the page does not invert. Given a cotangent of 1.7321, the angle could be thirty degrees, two hundred and ten degrees, or any of the infinitely many angles a half turn apart, and choosing between them needs a decision this page does not make.

Frequently asked questions

Why does the cotangent have no value at 0 degrees?
Because the cotangent is the adjacent side divided by the opposite side, and at zero degrees the opposite side has shrunk to nothing. Divide by zero and there is no answer. It is the same failure the tangent has at ninety degrees, moved a quarter turn: the two functions are reciprocals, so wherever one is zero the other is undefined, and the zeros and the poles change places.
Is the cotangent just one over the tangent?
Yes, and that is the shortest description of it. It is also the cosine divided by the sine, and the adjacent side over the opposite side, and all three say the same thing. The one place to be careful is at the angles where the tangent is zero or undefined: one over zero is not a number, so the cotangent is undefined at zero and one hundred and eighty degrees even though the tangent is defined there, and it is zero at ninety and two hundred and seventy degrees where the tangent is not.
Why is the cotangent of 90 degrees zero?
Because at a right angle the adjacent side has shrunk to nothing while the opposite side is at its longest. A ratio with zero on top is zero, so the reading is an ordinary zero — nothing is being special-cased. This is the angle the trigonometry page refuses, and the contrast is the point: the two functions swap which of them has an answer and which does not.
What does a negative cotangent mean?
That the two legs of the triangle have opposite signs, which happens in the second and fourth quadrants. Past a right angle the adjacent side points the other way while the opposite side does not, so the ratio turns negative and stays that way until one hundred and eighty degrees, turns positive again in the third quadrant where both are negative, and turns negative once more past two hundred and seventy.
Does the cotangent repeat, and how often?
Every half turn, which is twice as often as the sine and the cosine. Adding one hundred and eighty degrees to an angle flips the sign of both the adjacent side and the opposite side, and a ratio of two flipped quantities is unchanged. So thirty degrees and two hundred and ten degrees share a cotangent of 1.7321, and the table below shows the pair on rows eight apart. The sine and the cosine, by contrast, need a whole turn to come back.
Can the angle be entered in radians?
Yes — the box has a unit selector beside it and the reading is the same angle either way. Forty-five degrees and 0.7854 radians give the same cotangent, one. As always, the thing to check is which unit a source you are comparing against assumes, because most tables are written in degrees.

References

Related calculators

Tangent CalculatorTangent of an angle in degrees or radians.Trigonometry CalculatorSine, cosine and tangent of any angle, in degrees or radians.Unit Circle CalculatorSine, cosine and tangent of an angle, read off the unit circle.Cosine CalculatorCosine of an angle in degrees or radians.Right Triangle CalculatorSolve a right triangle from two sides or a side and angle.