Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Match each expression in Column I with its value in Column \(I I\). A. \(\frac{1}{2}\) B. \(\frac{\sqrt{2}}{2}\) c. \(\frac{\sqrt{3}}{2}\) D. \(-\sqrt{3}\) E. \(\frac{\sqrt{3}}{3}\) F. \(\sqrt{3}\) $$\frac{2 \tan \frac{\pi}{3}}{1-\tan ^{2} \frac{\pi}{3}}$$

Step-by-Step Solution

Answer

The value of the expression is \(-\sqrt{3}\), which matches D.

Step 1: Identify Values of Trigonometric Functions

First, identify the value of \( \tan \frac{\pi}{3} \). From trigonometric tables, we know that \( \tan \frac{\pi}{3} = \sqrt{3} \).

Step 2: Substitute into Given Expression

Substitute \( \tan \frac{\pi}{3} = \sqrt{3} \) into the expression. We have \( \frac{2 \tan \frac{\pi}{3}}{1 - \tan^2 \frac{\pi}{3}} = \frac{2 \times \sqrt{3}}{1 - (\sqrt{3})^2} \).

Step 3: Simplify the Denominator

Simplify the denominator \(1 - (\sqrt{3})^2 = 1 - 3 = -2\).

Step 4: Compute Expression

Now, the expression in the numerator is \(2 \times \sqrt{3} = 2\sqrt{3}\). Substitute this and the simplified denominator into the original expression: \(\frac{2\sqrt{3}}{-2}\).

Step 5: Final Calculation

Simplify the fraction \(\frac{2\sqrt{3}}{-2} = -\sqrt{3}\). This corresponds to option D from Column I.

Key Concepts

Trigonometric Values

Understanding trigonometric values is essential when working with trigonometric functions. These are often derived from the unit circle, which is a powerful tool in trigonometry. For example, knowing that the tangent of an angle is defined as the ratio of the sine and cosine functions at that angle is useful. You need to be familiar with common trigonometric values, especially for special angles like \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \text{and} \frac{\pi}{2} \).
- For \( \frac{\pi}{6} \): - \( \sin \frac{\pi}{6} = \frac{1}{2} \) - \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) - \( \tan \frac{\pi}{6} = \frac{\sqrt{3}}{3} \)
- For \( \frac{\pi}{4} \): - \( \sin \frac{\pi}{4} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) - \( \tan \frac{\pi}{4} = 1 \)
- For \( \frac{\pi}{3} \): - \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \) - \( \cos \frac{\pi}{3} = \frac{1}{2} \) - \( \tan \frac{\pi}{3} = \sqrt{3} \)
Understanding these values will assist you in evaluating expressions like the one in the original exercise.

Simplifying Expressions

Simplifying mathematical expressions is a crucial skill that makes complex problems easier to solve. When simplifying expressions, re-evaluate each component and apply basic algebraic rules and identities effectively. Let's break down simplification into steps using the example in the exercise:
1. **Substitution**: Start by replacing variables or other placeholders with their corresponding known values. This provides a clearer view of the expression. Example: Substitute \( \tan \frac{\pi}{3} = \sqrt{3} \) into the expression.2. **Simplify the Denominator**: Sometimes the denominator makes an expression difficult to evaluate. Simplifying it can be achieved by combining like terms or applying identities. In the exercise, simplify \(1 - (\sqrt{3})^2 \) to \(-2\).3. **Fraction Reduction**: Once the numerator and denominator are simplified, reduce the fraction. Divide both the numerator and the denominator by their greatest common divisor if possible. For instance, \( \frac{2\sqrt{3}}{-2} \) reduces to \(-\sqrt{3}\).By following these steps and approaching each component systematically, you can simplify even the most daunting trigonometric expressions efficiently.

Tangent Function

The tangent function is one of the primary trigonometric functions, alongside sine and cosine. It is essential to understand both its definition and its behavior, especially around specific angles. In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side, or on the unit circle, it's the sine of the angle divided by the cosine.
- **Key Definition**: - \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)The tangent function has a unique property where it becomes undefined whenever the cosine equals zero, such as at \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots \) because division by zero is undefined. Knowledge of these points is crucial when analyzing the function’s domain and range.
- For instance, as in the exercise with \( \tan \frac{\pi}{3} = \sqrt{3} \), understanding these properties helps quickly identify the trigonometric values needed for calculation.The tangent function is periodic with a period of \( \pi \), meaning it repeats its values every \( \pi \) units. Recognizing these traits in the tangent function helps in solving problems involving it, such as finding its values at various points or simplifying expressions involving tangents.