Problem 78 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically. $$\frac{1}{2}\left(\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}\right)$$

Step-by-Step Solution

Answer

The trigonometric function equivalent to the given expression is sec \(\theta\).

Step 1: Simplify the Expression

Combine the terms by getting a common denominator: \[\frac{1}{2}\left(\frac{1+\sin \theta}{\cos \theta}+\frac{\cos \theta}{1+\sin \theta}\right)= \frac{1}{2}\left(\frac{(1+\sin \theta)^2 + \cos^2 \theta}{\cos \theta(1+\sin \theta)}\right)\]

Step 2: Apply the Pythagorean identity

Apply the Pythagorean identity \(\sin^2 \theta+ \cos^2 \theta=1\) to the numerator to further simplify the expression:\[\frac{1}{2}\left(\frac{(1+\sin \theta)^2 + \cos^2 \theta}{\cos \theta(1+\sin \theta)}\right) = \frac{1}{2}\left(\frac{1 + 2\sin \theta + 1}{\cos \theta(1+\sin \theta)}\right) =\frac{1}{\cos \theta}\]

Step 3: Identify the Trigonometric Function

Relate the final simplified expression to its corresponding trigonometric function. In this case, the expression simplifies to \(\frac{1}{\cos \theta}\), which is equivalent to sec \(\theta\)

Step 4: Verify the Algebraic Correction

Verify whether the original expression and the identified trigonometric function yield the same result for different values of theta. If the results tally, then the identified function is correct.

Key Concepts

Pythagorean Identity

The Pythagorean identity is an essential part of trigonometry, forming the foundation of many trigonometric proofs. It stems from the Pythagorean theorem and is given by the equation:

• \[\sin^2 \theta + \cos^2 \theta = 1\]

This equation tells us that the square of the sine of an angle plus the square of the cosine of the same angle is always equal to one. This identity helps us transform expressions involving \( \sin \theta \) and \( \cos \theta \).
In our exercise, this identity allowed simplifying the expression in the numerator by replacing one of its terms, \( \cos^2 \theta \), with \( \sin^2 \theta + \cos^2 \theta = 1 \), and then further simplifying the expression to find its equivalent trigonometric function.

Simplifying Expressions

Simplifying expressions is a critical skill in mathematics that reduces complex expressions into simpler, equivalent forms. Simplification often involves:

• Finding a common denominator to combine fractions.
• Using trigonometric identities like the Pythagorean identity.
• Applying basic algebraic operations to consolidate terms.

In the given exercise, the simplification process began by combining the terms under a common denominator, \( \cos \theta(1+\sin \theta) \). This made it easier to manipulate and eventually relate the expression to a known trigonometric function. By simplifying complex terms, it becomes possible to accurately determine equivalent expressions or functions.

Graphing Utility

Graphing utilities, like graphing calculators or software, are excellent tools for visualizing and solving trigonometric problems. They allow you to:

• Plot trigonometric functions and visualize their behavior across different angles.
• Verify algebraic solutions by comparing them with plotted graphs.
• Experiment with different values to observe changes and patterns.

In the context of the exercise, using a graphing utility can help verify that the simplified expression, \( \frac{1}{\cos \theta} \), corresponds to the secant function. By graphing both the original expression and the secant function, we can confirm their equivalence visually, adding an additional layer of verification beyond algebraic simplification.

Secant Function

The secant function, denoted as \( \sec \theta \), is a trigonometric function defined as:

• \[ \sec \theta = \frac{1}{\cos \theta} \]

It is the reciprocal of the cosine function and is essential in various trigonometric calculations, especially when dealing with ratios of sides in right triangles. Secant is particularly useful when division by zero must be avoided since it approaches infinity when \( \cos \theta = 0 \).
In our exercise, the expression \( \frac{1}{\cos \theta} \) was simplified to show that it is equal to the secant function. Understanding this relationship allows us to reframe complex expressions in terms of more familiar trigonometric functions, making them easier to work with over different domains of \( \theta \).