Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the fundamental identities to simplify the expression. Use the table feature of a graphing utility to check your result numerically. $$\frac{\csc x}{\cot x}$$

Step-by-Step Solution

Answer

\(\frac{\csc x}{\cot x}\) simplifies to \(\sec x\).

Step 1: Recall Fundamental Trigonometric Identities

The fundamental identities you'll need to know are: \(\csc x = \frac{1}{\sin x}\) (definition of cosecant) and \(\cot x = \frac{\cos x}{\sin x}\) (definition of cotangent).

Step 2: Substitute Identifiers

Substitute \(\csc x\) and \(\cot x\) with their definitions. This turns the expression into \(\frac{\frac{1}{\sin x}}{\frac{\cos x}{\sin x}}\).

Step 3: Simplify the Expression

The expression can be simplified to \(\frac{1}{\cos x}\).

Step 4: Apply the Trigonometric Identity

Substitute \(\frac{1}{\cos x}\) with its trigonometric identity. The simplified expression is therefore \(\sec x\).

Step 5: Verify the Result

Use the table feature of a graphing utility to verify the result. The values of \(\frac{\csc x}{\cot x}\) and \(\sec x\) should match for any input \(x\).

Key Concepts

Simplifying Trigonometric Expressions

When simplifying trigonometric expressions, it is important to use fundamental identities to make complicated functions more manageable. One common simplification technique is converting complex trigonometric functions into simpler ones by substituting them with their basic identities.
For example, when given an expression such as \( \frac{\csc x}{\cot x} \), it's useful to recall some basic trigonometric identities:

• \( \csc x = \frac{1}{\sin x} \)
• \( \cot x = \frac{\cos x}{\sin x} \)

By substituting these identities into the original expression, we can transform a seemingly complex expression into something simpler, helping to avoid confusion and errors in further calculations.
Simplifying is essentially breaking down the trigonometric expression to its core components, which, in this case, results in replacing the complicated fraction with the single function \( \sec x \). This assists in solving equations more easily and efficiently, and it's a commonly used skill in calculus and other advanced math subjects.

Cosecant and Cotangent

Cosecant and cotangent are reciprocal trigonometric functions. Understanding these functions is vital for simplifying trigonometric expressions and can be quite useful in solving various mathematical problems.
Cosecant Function:

• Defined as \( \csc x = \frac{1}{\sin x} \), which means it is the reciprocal of the sine function.
• Cosecant provides a way to express angles and sides in terms of ratios, particularly in trigonometric problems involving triangles.

Cotangent Function:

• Defined as \( \cot x = \frac{\cos x}{\sin x} \), making it the reciprocal of the tangent function.
• Cotangent is useful in problems where we relate angles to the opposite and adjacent sides in right triangles.

These functions are often used in combination with other functions and identities to simplify expressions or solve equations. Each has its own set of properties and behaviors, which are vital to understanding when working with trigonometric expressions and identities.

Graphing Utility Verification

Graphing utilities are incredibly useful tools for verifying mathematical results, particularly with trigonometric expressions. They allow for visual confirmation that expressions have been simplified correctly and behave as expected over specific domains.
After simplifying a trigonometric expression, like turning \( \frac{\csc x}{\cot x} \) into \( \sec x \), using a graphing utility can reassure you that the simplification is accurate:

• Input the original expression and the simplified version into the graphing utility.
• Check the table feature to compare numerical values for different values of \( x \).
• If the values match for each \( x \), you can confidently assert that the simplification is correct.

This approach not only helps confirm the accuracy of mathematical work but also aids in developing a deeper intuitive understanding of trigonometric functions and their identities. Verifying with technology reinforces the learning and gives confidence in applying these skills to more complex problems in the future.