Problem 74 - Precalculus with Limits

Question

In Exercises 71 - 74, simplify the expression algebraically and use a graphing utility to confirm your answer graphically. \( \tan(\pi + \theta) \)

Step-by-Step Solution

Answer

\( \tan(\pi + \theta) \) simplifies to \( \tan(\theta) \)

Step 1: Use the periodicity properties of the tangent function

The tangent function has a period of \( \pi \), which means \( \tan(\pi + \theta) \) = \( \tan(\theta) \). This holds true because adding or subtracting any multiple of \( \pi \) to the argument of the tangent function does not change its value. Therefore, \( \tan(\pi + \theta) \) simplifies to \( \tan(\theta) \).

Step 2: Graphical confirmation

To confirm the solution graphically, one must plot on a graphing utility both the initial expression \( \tan(\pi + \theta) \) and the obtained result \( \tan(\theta) \). Both expressions should produce identical graphs, confirming algebraic simplification is correct.

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan(\theta) \), is one of the basic trigonometric functions alongside sine and cosine. It is defined as the ratio of the sine function to the cosine function: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This function is particularly interesting because it is undefined wherever the cosine function is zero, leading to vertical asymptotes on its graph. This usually occurs at odd multiples of \( \frac{\pi}{2} \).

• The graph of the tangent function extends infinitely in both directions, creating a series of repeating wave-like patterns.
• Its range is all real numbers, meaning there is no upper or lower limit to the values it can take.
• The tangent function crosses the origin at the point \( (0,0) \) and repeats its pattern every \( \pi \) units horizontally.

Understanding these properties helps in visualizing how the function behaves and why it simplifies as it does in certain conditions.

Periodicity

Periodicity is a key characteristic of trigonometric functions. For the tangent function, this is expressed by its period of \( \pi \). This means that the tangent function repeats its values every \( \pi \) radians.

• When the problem states \( \tan(\pi + \theta) \), it's pointing out that the function will return to the same value as \( \tan(\theta) \).
• This leads to the simplification that the two expressions, \( \tan(\pi + \theta) \) and \( \tan(\theta) \), are actually equivalent.

By understanding this, you see that shifting the input of the function by \( \pi \) radians does not affect its value, emphasizing the periodic nature of the tangent function. This is an essential concept that confirms \( \tan(\pi + \theta) = \tan(\theta) \), demonstrating that the expressions are essentially the same.

Graphical Confirmation

Graphical confirmation is a practical approach to verify algebraic solutions, especially for trigonometric identities. Using a graphing utility, we can visualize the tangent function's behavior and observe its periodicity.

• To confirm \( \tan(\pi + \theta) = \tan(\theta) \), plot both functions on the same graph.
• If the graphs overlap perfectly, it visually confirms that the algebraic simplification is correct.
• This approach provides a clear visual understanding of why the expressions are identical over the range of the function's period.

Graphical visualization can be an invaluable tool in learning, offering intuitive insights that complement analytical results. It helps to reinforce the concept of periodicity and the unique characteristics of trigonometric functions.