Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. $$\cot \theta, \text { given that } \tan \theta=5$$

Step-by-Step Solution

Answer

\( \cot \theta = \frac{1}{5} \)

Step 1: Recall the reciprocal identity

The reciprocal identity states that \( \cot \theta = \frac{1}{\tan \theta} \). This identity will be utilized to find \( \cot \theta \).

Step 2: Apply the reciprocal identity

Based on the given \( \tan \theta = 5 \), use the reciprocal identity to calculate \( \cot \theta \):\[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{5} \]

Step 3: Rationalize the denominator (if needed)

Since \( \frac{1}{5} \) is already a rationalized fraction, no further changes are required. The value of \( \cot \theta \) in this case is already in its simplest form.

Key Concepts

Reciprocal Identity

In trigonometry, reciprocal identities are fundamental for converting between various trigonometric functions. A reciprocal identity helps us express one trigonometric function in terms of another. This can be very handy when you know one function's value and need to find another function's value.
For instance, when you have the tangent of an angle, \(\tan \theta\), the reciprocal identity allows you to find the cotangent, \(\cot \theta\). The relationship between them is given by:

• \(\cot \theta = \frac{1}{\tan \theta}\)

This linkage between cotangent and tangent shows that they are reciprocal to each other. Understanding and using these reciprocal relationships can simplify solving trigonometric problems, allowing for more elegant solutions and deeper insights into the functions' behavior.

Cotangent Function

The cotangent function, \(\cot \theta\), is one of the six main trigonometric functions. It is the reciprocal of the tangent function. This means if you know the tangent, you can easily find the cotangent by finding its reciprocal. The formula is simple:

• \(\cot \theta = \frac{1}{\tan \theta}\)

The cotangent function itself measures the ratio of the adjacent side to the opposite side in the context of a right triangle. While not as commonly used as sine or cosine, cotangent is crucial in certain identities and trigonometric equations. By understanding cotangent's reciprocal nature, solving problems involving angles and triangles becomes more straightforward.
When working with the cotangent function, particularly in analytical contexts, values are often presented in their simplest forms. This keeps calculations neat and reduces potential errors, offering clear insights into geometric relationships.

Rationalizing Denominators

Rationalizing denominators is an important mathematical process that simplifies expressions, making them easy to understand and use. When dealing with fractions, especially those involving square roots or irrational numbers, it's often required to adjust them so the denominator becomes a rational number.
This process ensures that denominators are free from any square roots or irrational numbers. For most cases involving trigonometric expressions, rationalizing helps maintain simplicity and clarity. Here's a basic rundown of the process:

• If you have a fraction like \(\frac{\sqrt{2}}{3}\), you multiply both the numerator and denominator by the conjugate of the denominator to eliminate the square root.

Although not needed in our example of \(\frac{1}{5}\), as 5 is already rational, understanding how to rationalize is key. It maintains expressions in a form that is often considered more organized or conventional in mathematics, allowing for easier comparison and further manipulation in problem-solving scenarios.