Problem 52 - Precalculus : Building Concepts and Connections

Question

Find a cofunction that has the same value as the given quantity. $$\cot 72^{\circ}$$

Step-by-Step Solution

Answer

The cofunction that has the same value as \(\cot(72^{\circ})\) is \(\tan(18^{\circ})\).

Step 1: Identify the cofunction identity for Cotangent

The problem can be solved by identifying the co-function identity of the cotangent function. We have the cofunction identity for Cotangent as \(\cot(A) = \tan(90^{\circ} - A)\).

Step 2: Substitute the value into the co-function identity

Substituting the given value into the cofunction identity, we get \(\cot(72^{\circ}) = \tan(90^{\circ} - 72^{\circ})\).

Step 3: Simplify the expression

Simplify the right side of the expression, we get \(\cot(72^{\circ}) = \tan(18^{\circ})\).

Key Concepts

Cotangent Function

Understanding the cotangent function is crucial in trigonometry. It's one of the six fundamental trigonometric functions. The cotangent function, often abbreviated as 'cot', is related to the tangent function but it's not as commonly discussed.

The cotangent of an angle in a right triangle is the ratio of the length of the adjacent side to the opposite side. In mathematical terms, if \theta is an angle, then the cotangent of \theta, written as \( \cot(\theta) \), is defined as \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \), where \( \cos \) and \( \sin \) are cosine and sine functions respectively.

Tangent Function

The tangent function is another important player in the world of trigonometry. This function, usually abbreviated as 'tan', is the ratio of the length of the opposite side to the adjacent side of a right triangle for a given angle.

When we talk about the tangent of an angle \( \theta \), denoted as \( \tan(\theta) \), we are referring to \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). Unlike the cotangent, which can be seen as 'adjacent over opposite', the tangent function conversely considers 'opposite over adjacent'.

The tangent function has a range of \( (-\infty, \infty) \) and is periodic with a period of \( \pi \) radians, or 180 degrees, which means it repeats its values every 180 degrees. It's important in solving many types of trigonometric problems, including those involving angles of elevation and depression.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved. They are fundamental tools in not just trigonometry, but in mathematics as a whole because they enable simplification and transformation of trigonometric expressions. The most common types include Pythagorean identities, reciprocal identities, and co-function identities.

Co-Function Identities

A particularly interesting subset of trigonometric identities is the co-function identities, which express the relationship between sine, cosine, tangent, cotangent, secant, and cosecant of complementary angles. For example, \( \sin(90^\circ - \theta) = \cos(\theta) \), \( \cos(90^\circ - \theta) = \sin(\theta) \), and, relevant to our exercise, \( \cot(90^\circ - \theta) = \tan(\theta) \). These identities are extremely useful in transforming expressions and solving equations that involve trigonometric functions.

Utilizing these identities, exercises like finding a cofunction that matches the value of \( \cot(72^\circ) \) become straightforward, as seen in the solution provided where \( \cot(72^\circ) \) was effectively transformed into \( \tan(18^\circ) \) using the co-function identity between cotangent and tangent.