Problem 87 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The quotient identities describes how the tangent of an angle is obtained by dividing the sine of that angle by its cosine, and similarly, the cotangent of an angle is obtained by dividing the cosine of the angle by its sine.

Step 1: Understanding the quotient identities

The two quotient identities in trigonometry are equations that establish the relationship between tangent and cotangent functions with the sine and cosine functions. The first identity defines tangent of a given angle as the ratio of the sine to the cosine of that angle, and the second identity defines the cotangent of an angle as the ratio of the cosine to the sine of that angle.

Step 2: Describing the identities using words

To describe this using words, we can say that in the first identity, when the length of the side opposite an angle (represented by sine) in a right triangle is divided by the length of the adjacent side (represented by cosine), the result is the tangent of that angle. For the second identity, if we divide the length of the adjacent side (represented by cosine) by the length of the side opposite (represented by sine), the result is the cotangent of that angle.

Step 3: Wrapping up

In conclusion, these quotient identities are ways to express the tangent and cotangent trigonometric functions in terms of the sine and cosine functions, effectively linking all four of these major trigonometric functions.