Problem 233 - Precalculus

Question

For the following exercises, use a graphing calculator to evaluate. $$ \tan \frac{5 \pi}{8} $$

Step-by-Step Solution

Answer

Using a calculator, \(\tan \frac{5\pi}{8}\) evaluates to approximately -2.414.

Step 1: Understand the Problem

We need to evaluate \(\tan \frac{5\pi}{8}\). This involves finding the tangent of an angle given in radians, which is \(\frac{5\pi}{8}\).

Step 2: Set Up the Calculator

Turn on your graphing calculator. Ensure it is in radian mode because the angle \(\frac{5\pi}{8}\) is given in radians.

Step 3: Input the Expression

Enter the expression into the calculator. Type \(\tan\left(\frac{5\pi}{8}\right)\). Use the \(\pi\) button on your calculator for accuracy.

Step 4: Evaluate and Record the Result

Press the 'Enter' or 'Evaluate' button on your calculator to compute the value. The calculator should display the result of the tangent function evaluated at \(\frac{5\pi}{8}\).

Key Concepts

Tangent Function

The tangent function, represented as \( \tan(\theta) \), is one of the primary trigonometric functions. It comes after the sine and cosine functions in the sequence of foundational trigonometric functions.

• The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
• Mathematically, it can be expressed as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
• Its value depends on both sine and cosine; if \( \cos(\theta) \) is zero, the tangent is undefined because it leads to a division by zero.

The tangent function is periodic with period \( \pi \), which means it repeats its values in intervals of \( \pi \) radians.
This function is particularly interesting as it can take any real value, from negative to positive infinity, depending on the angle \( \theta \).
The graph of the tangent function is characterized by its vertical asymptotes, where the function tends to infinity as \( \cos(\theta) \) approaches zero.

Radian Measure

Radian measure is a way of representing angles based on the radius of a circle. This system is often used in calculus and trigonometry because it simplifies many mathematical formulas.

• An angle measured in radians corresponds to the length of the arc that it subtends in a circle whose radius is one unit.
• A full circle is \( 2\pi \) radians, which is equivalent to 360 degrees.
• Thus, \( \pi \) radians equal 180 degrees, and smaller angles can be easily calculated from these correlations.

Using radians can make analyzing trigonometric functions more straightforward, as the unit circle—often used in trigonometry—naturally incorporates radian measurement.
When an exercise presents an angle such as \( \frac{5\pi}{8} \), it suggests an angle partway between \( \frac{\pi}{2} \) (or 90 degrees) and \( \pi \) (or 180 degrees).
Understanding radian measure is crucial for solving problems in physics, engineering, and other fields where angles are a fundamental part of the analysis.

Graphing Calculator Usage

A graphing calculator is a powerful tool that aids in visualizing and evaluating mathematical expressions, especially in areas such as trigonometry. Here's how you can use a graphing calculator effectively for trigonometric evaluations:

• Radian Mode: Ensure the calculator is in 'radian mode' when dealing with angles in radians, like \( \frac{5\pi}{8} \). This setting is crucial for accurate calculations related to angle measures that aren't in degrees.
• Input Functions: Familiarize yourself with the calculator's interface for inputting functions. Most graphing calculators have a \( \pi \) button which allows for precise angle calculations without the need to approximate \( \pi \).
• Function Syntax: Use the function buttons or respective syntax to input expressions like \( \tan\left(\frac{5\pi}{8}\right) \). This ensures the calculator treats the input as a mathematical expression to be evaluated.
• Results Interpretation: Upon calculation, interpret the results shown. The graphing calculator might provide the tangent value directly, or you may need to locate the relevant section of a graph.

Knowing how to navigate and use these devices efficiently can greatly enhance your understanding and accuracy in solving trigonometric problems, making the process smoother and more intuitive.