Problem 34 - 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. $$\cos \beta \tan \beta$$

Step-by-Step Solution

Answer

The simplified form of \( \cos \beta \tan \beta \) is \( \sin \beta \)

Step 1: Apply Fundamental Identities

To simplify the expression \( \cos \beta \tan \beta \), first recognize that \( \tan \beta \) is equivalent to \( \frac{\sin \beta}{\cos \beta} \) according to the trigonometric identities. Then, substitute \( \tan \beta \) with \( \frac{\sin \beta}{\cos \beta} \), you will obtain \( \cos \beta \cdot \frac{\sin \beta}{\cos \beta} \)

Step 2: Simplify the Expression

Next, you can cancel out \( \cos \beta \) in the numerator and the denominator to simplify the expression, and you will end up with \( \sin \beta \).

Step 3: Numerical Verification

Lastly, use the table feature of a graphing utility to confirm the numerical equivalence of \( \cos \beta \tan \beta \) and \( \sin \beta \) for a few values of \( \beta \). For instance, compare the values for \( \beta = 1, 2, 3, \ldots \)

Key Concepts

Fundamental Trigonometric Identities

Understanding the fundamental trigonometric identities is crucial for simplifying trigonometric expressions. These identities are equalities involving trigonometric functions that hold for all values of the involved angles, where they are defined. Among the most important ones are:

• The Pythagorean identities, like \( \text{sin}^2(x) + \text{cos}^2(x) = 1 \).
• The reciprocal identities, such as \( \text{sec}(x) = \frac{1}{\text{cos}(x)} \) and \( \text{csc}(x) = \frac{1}{\text{sin}(x)} \).
• The quotient identities, which express \( \text{tan}(x) = \frac{\text{sin}(x)}{\text{cos}(x)} \) and \( \text{cot}(x) = \frac{\text{cos}(x)}{\text{sin}(x)} \).
• The co-function identities, like \( \text{sin}(90° - x) = \text{cos}(x) \) and vice versa.

In our exercise, the quotient identity is used to express \( \text{tan}(\beta) \) in terms of \( \text{sin}(\beta) \) and \( \text{cos}(\beta) \), which allows for the simplification of the original expression \( \text{cos}(\beta) \text{tan}(\beta) \). Remember, practicing these identities and knowing when and how to use them will help you tackle trigonometric simplifications with ease.

Graphing Utility

A graphing utility is an incredibly helpful tool for visualizing and verifying trigonometric expressions. It can take various forms, such as a graphing calculator, specialized software, or even online applications. When dealing with complex trigonometric formulas or when aiming to verify the results of simplification, graphing utilities offer the following benefits:

• Visualization: By plotting the functions, one can see how they behave and confirm their equivalence visually.
• Numerical Evaluation: These tools often have table features that list the values of functions at specific angles, enabling quick numerical comparisons.

For instance, after simplifying \( \text{cos}(\beta) \text{tan}(\beta) \) to \( \text{sin}(\beta) \), using the table feature can confirm that at the same angle \( \beta \), both expressions yield the same numerical value. This aids in ensuring that the simplification is indeed correct.

Numerical Verification

Numerical verification is the process of checking the correctness of mathematical expressions by evaluating their values at specific points. It serves as an additional check to ensure our algebraic manipulation is on point. In our context of simplifying trigonometric expressions, numerical verification helps in the following ways:

• Validates the logic used in simplification by comparing the original expression and the simplified result.
• Acts as a safeguard against common mistakes such as identity misapplication or algebraic errors.
• Boosts confidence in the result before moving on to real-world applications or higher-level math problems.

In the given exercise, this approach would involve plugging in values for \( \beta \), such as \( \beta = 1, 2, 3, \) and ensuring that both \( \text{cos}(\beta) \text{tan}(\beta) \) and the simplified result \( \text{sin}(\beta) \) match numerically. If not, one might have to revisit the steps to find out where the mistake occurred.