Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Write expression in terms of sine and cosine, and simplify it. (The final expression does not have to be in terms of sine and cosine.) $$\cos \theta \csc \theta$$

Step-by-Step Solution

Answer

The simplified expression is \( \cot \theta \).

Step 1: Write the Given Expression

The given expression is \( \cos \theta \csc \theta \).

Step 2: Convert Cosecant to Reciprocal Identity

The cosecant function \( \csc \theta \) is the reciprocal of the sine function. Thus, we can write \( \csc \theta = \frac{1}{\sin \theta} \).

Step 3: Substitute the Reciprocal Identity into the Expression

Replace the \( \csc \theta \) in the original expression with \( \frac{1}{\sin \theta} \). The expression becomes \( \cos \theta \times \frac{1}{\sin \theta} \).

Step 4: Simplify the Expression

Multiply the terms resulting in \( \frac{\cos \theta}{\sin \theta} \).

Step 5: Recognize Cotangent Identity

The expression \( \frac{\cos \theta}{\sin \theta} \) is equivalent to the cotangent function, \( \cot \theta \).

Key Concepts

Cosecant: Understanding the Reciprocal

The cosecant function, denoted as \( \csc \theta \), is one of the lesser-known trigonometric functions. It is the reciprocal of the sine function. In simpler terms, if you know the sine of an angle, you can easily find its cosecant. To find \( \csc \theta \), just calculate \( \frac{1}{\sin \theta} \). Here are some key points to remember:

• Think of cosecant whenever you see sine and its reciprocal is needed.
• Cosecant is undefined wherever sine equals zero since division by zero is not possible.

This function might not be the most mainstream, but it comes in handy, especially in simplifying expressions more complex than they first appear.

Cotangent: The Quotient of Cosine and Sine

Cotangent, denoted as \( \cot \theta \), is another interesting trigonometric function. It represents the reciprocal of the tangent function. Essentially, it’s the ratio of the adjacent side to the opposite side in the right triangle, just reversed from what tangent represents. Mathematically, cotangent can be expressed as \( \frac{\cos \theta}{\sin \theta} \). Key things to note about cotangent include:

• It can also be viewed as a measure of the difference in "steepness" between sine and cosine.
• Cotangent is undefined when sine is zero because division by zero is undefined.

Understanding cotangent is crucial for solving many trigonometry problems, particularly those involving complementary angles.

Reciprocal Identities: A Quick Guide

Reciprocal identities form a fundamental concept in trigonometry. They show how any trigonometric function can be expressed as the reciprocal of another. These identities are very useful for simplifying expressions and solving equations. For example, three primary reciprocal identities are:

• \( \csc \theta = \frac{1}{\sin \theta} \)
• \( \sec \theta = \frac{1}{\cos \theta} \)
• \( \cot \theta = \frac{1}{\tan \theta} \)

These identities help transform complex expressions into simpler forms, making them easier to work with. Keep these identities handy when simplifying trigonometric expressions or solving complex trigonometry problems.

Simplification in Trigonometry: Steps to Success

Simplifying trigonometric expressions can make solving problems much easier. It often involves converting functions to terms of sine and cosine, using reciprocal identities, and recognizing equivalent expressions. Here’s a quick guide to successful simplification:

• Convert all functions to sine and cosine using known identities. This forms a common ground for comparison.
• Look for opportunities to apply reciprocal identities. These often transform complex expressions into simpler forms.
• Identify and use known ratios, like the fact that \( \frac{\cos \theta}{\sin \theta} \) is several times cotangent, defined as \( \cot \theta \).
• Simplify algebraically to condense the expression as much as possible.

This process turns a potentially daunting problem into a more manageable one, providing clearer pathways to solutions.