Problem 70 - A Graphical Approach to Precalculus with Limits

Question

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. $$\cot \theta, \text { given that } \tan \theta=18$$

Step-by-Step Solution

Answer

\( \cot \theta = \frac{1}{18} \)

Step 1: Understand Reciprocal Identity

The reciprocal identity for tangent is given by \( \cot \theta = \frac{1}{\tan \theta} \). This tells us that cotangent is the reciprocal of tangent.

Step 2: Apply the Reciprocal Identity

Given that \( \tan \theta = 18 \), we can find \( \cot \theta \) by using the reciprocal identity: \[ \cot \theta = \frac{1}{18} \].

Step 3: Result Simplification

Since \( \frac{1}{18} \) is already a simplified rational number and there is no denominator to rationalize further, we conclude that \( \cot \theta = \frac{1}{18} \). This is our final answer.

Key Concepts

Reciprocal Identity

Reciprocal identities are key tools in trigonometry. They help us express one trigonometric function in terms of another. For tangent and cotangent, these identities show how they relate to each other. Specifically, the reciprocal identity states that the cotangent function is the reciprocal of the tangent function, meaning it can be expressed as:\[\cot \theta = \frac{1}{\tan \theta}\]Using reciprocal identities, we can solve problems where we know one function value and need another. This is particularly useful when working with trigonometric equations. Understanding this relationship is crucial for solving equations like the one in the exercise. By knowing the value of one function, you can quickly find its reciprocal function.

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is one of the six fundamental trigonometric functions. It relates to the angle in a right triangle and is defined as the ratio of the adjacent side to the opposite side. However, it is more commonly expressed using the reciprocal of the tangent:\[\cot \theta = \frac{1}{\tan \theta}\]The cotangent function becomes particularly useful when solving trigonometric problems that require inverses of tangent values. This function does not have a direct presence, like sine or cosine, but partners with tangent to bring more versatility into trigonometric calculations. The value of \( \cot \theta \) is indefinite at angles where tangent equals zero, which need special attention.

Tangent Function

The tangent function, symbolized as \( \tan \theta \), is a fundamental concept in trigonometry. It is defined as the ratio of the opposite side to the adjacent side in a right triangle:\[\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\]This function is vital for understanding and calculating angles and sides in trigonometry. Its values repeat in a cyclical pattern, making it a periodic function with a period of \( \pi \) radians. This cyclic nature means its values also include both positive and negative numbers.While sometimes overshadowed by sine and cosine, the tangent function is equally indispensable. Its counterpart, cotangent, relies directly on its values, as shown in our reciprocal identity. When you comprehend how \( \tan \theta = 18 \) leads directly to \( \cot \theta = \frac{1}{18} \), you see them working in tandem to solve trigonometric problems.