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The Cotangent Function

The cotangent function, abbreviated cot, is called a reciprocal function because it uses the reciprocal of the trigonometry function, tangent (hence, the co-). The cotangent function uses the ratio


So if you look at the following figure, you see that the two cotangents are



The ratio for the cotangent is just the ratio, so the numbers in the ratio are not necessarily the lengths of the sides. The fraction made by the lengths might’ve been reduced by dividing the numerator and denominator by the same number.

Sometimes you know the value of the cotangent along with other info and have to solve for one or both of the sides. Try this example: What are the lengths of the legs of a right triangle if


and the hypotenuse is 183 inches long?

  1. Write the adjacent and opposite sides as multiples of the same number, m, and put them in the Pythagorean theorem with the hypotenuse.

    (11m)2 + (60m)2 = 1832

  2. Simplify the equation and solve for m.

  3. Use the value of m to find the lengths of the two legs.

    Because you know that m = 3, you know that the adjacent side is 11m = 11 3 = 33 and the opposite side is 60m = 60 3 = 180. The three sides of the right triangle are 33, 180, and 183. You can double-check your results by plugging these three numbers into the Pythagorean theorem and making sure the theorem holds true.

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