Problem 86 - Algebra and Trigonometry

Question

Use words (not an equation) to describe one of the reciprocal identities.

Step-by-Step Solution

Answer

The reciprocal identity of sine, namely the cosecant, tells us that the ratio of the length of the hypotenuse to the length of the side opposite the angle in a right triangle is actually the reciprocal of the sine of that angle.

Step 1: Select a Reciprocal Identity from Trigonometry

There are three key reciprocal identities in trigonometry tied to the three key trigonometric functions - sine, cosine, and tangent. These are cosecant (csc), secant (sec), and cotangent (cot) respectively. Each of these is the reciprocal of one of the basic functions. For the purpose of this exercise, one identity will be picked - the reciprocal identity for sine, namely cosecant.

Step 2: Understand the Reciprocal Relationship

The reciprocal identity for sine, which is cosecant, tells us that the cosecant of an angle is actually the reciprocal of the sine of that same angle.

Step 3: Describe the Reciprocal Identity in Words

The reciprocal identity for the sine function can be described in words as such: The value of the cosecant of an angle in a right triangle is the ratio of the length of the hypotenuse to the length of the side opposite that angle. Importantly, this is the reciprocal of the sine function, which is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Therefore, the cosecant of an angle is effectively the reciprocal of the sine of that same angle.