The Cosecant Function

The cosecant function, abbreviated csc, is called a reciprocal function because it is the reciprocal of one of the three basic trig functions, sine. The cosecant function uses the ratio

The hypotenuse of a right triangle is always the longest side, so, when working with triangles, the numerator of this fraction is always larger than the denominator. As a result, the cosecant function always produces values bigger than 1.

You can use the values in the preceding figure to determine the cosecants of the two acute angles:

Suppose someone asks you to find the cosecant of angle if you know that the hypotenuse is 1 unit long and that the right triangle is isosceles. Remember that an isosceles triangle has two congruent sides. These two sides have to be the two legs, because the hypotenuse has to have the longest side. So to find the cosecant, you would follow these steps:

1. Find the lengths of the two legs.

   The Pythagorean theorem says that a² + b² = c², but because two sides are congruent, you can take out one variable and write the equation as a² + a² = c². Put in 1 for c and solve for a.

   

   You can leave the radical in the denominator and not worry about rationalizing, because you’re going to input the whole thing into the cosecant ratio anyway, and things can change.

2. Use the length of the opposite side in the ratio for cosecant.

   

The square root of 2 is about 1.4, so it fits right in there with the possible values of the cosecant of an angle.