Problem 236 - Precalculus

Question

For the following exercises, use a graphing calculator to evaluate. $$ \tan 98^{\circ} $$

Step-by-Step Solution

Answer

\( \tan 98^{\circ} \approx -5.6713 \)

Step 1: Understand the Task

We are tasked with finding the value of \( \tan 98^{\circ} \) using a graphing calculator. The tangent function, denoted as \( \tan \), is a trigonometric function that relates the angle to the ratio of the opposite side to the adjacent side in a right-angled triangle. However, for angles more than \( 90^{\circ} \), it's also interpreted on the unit circle.

Step 2: Set the Calculator to Degree Mode

Ensure that your graphing calculator is set to degree mode since the angle provided \( 98^{\circ} \) is in degrees. Most calculators have a setting option where you can switch between radians and degrees.

Step 3: Input the Angle Into the Calculator

On your graphing calculator, enter the function "tan" followed by the angle 98. You should press the TAN button and then type in "98". Make sure you have done this correctly and that no other functions are being used.

Step 4: Evaluate the Expression

Press the ENTER button to compute the tangent of \( 98^{\circ} \). The calculator will display the decimal value of \( \tan 98^{\circ} \).

Step 5: Record the Result

Observe and note down the result displayed on the calculator. This is the value of the tangent of \( 98^{\circ} \). The result should be approximately \(-5.6713\).

Key Concepts

Graphing Calculator

A graphing calculator is a versatile tool that many students use to handle complex calculations, particularly when it comes to trigonometry. With a graphing calculator, you can visually interpret functions like those involving sine, cosine, and tangent, which can be extremely helpful for understanding these mathematical concepts.

To use a graphing calculator for trigonometric functions, such as finding the value of \( \tan 98^{\circ} \), follow these simple steps:

• Turn on the graphing calculator and open the trigonometric function menu.
• Select the function you need; in this case, "tan" for tangent.
• Input the angle you want to evaluate; ensure you are inputting the correct angle given in the problem.
• Press the calculate or enter button to receive the result.

Graphing calculators often have additional settings that allow you to visualize the function's graph. This feature is especially useful for understanding how the trigonometric functions behave as you input different angles.

Degree Mode

When working with angles, especially in trigonometric problems, it's crucial to ensure your graphing calculator is set to the correct mode. For angles measured in degrees, like \( 98^{\circ} \), you need to switch your calculator to degree mode rather than radian mode. This ensures accurate results.

To set your calculator to degree mode:

• Look for the "mode" setting on your calculator. This is often found by pressing the "Mode" button or by accessing the settings menu.
• Navigate through options until you locate the angular measurement modes. Select "Degrees" or "Deg" from the list.
• Confirm the selection before proceeding with calculations.

Choosing the correct mode is essential because different modes interpret angle measurements differently, impacting the outcomes of your calculations. For example, on most calculators, 180 degrees equal \( \pi \) radians, making it vital to pick the right setting based on your exercise requirements.

Unit Circle

The unit circle is a powerful tool in trigonometry that provides a graphical way to understand angles and their corresponding trigonometric values. The circle has a radius of 1 and is centered at the origin of the coordinate plane.

Here's how the unit circle helps with trigonometric functions like tangent:

• Every point on the unit circle corresponds to an angle measured from the positive x-axis, moving counterclockwise.
• The x-coordinate of a point on the unit circle represents the cosine of the angle, while the y-coordinate represents the sine.
• The tangent of the angle is the ratio of the y-coordinate to the x-coordinate (sine divided by cosine).
• For angles greater than \( 90^{\circ} \), the tangent value may become negative, which occurs when both sine and cosine have opposite signs.

To find the tangent of \( 98^{\circ} \), you can visualize this on the unit circle. This particular angle falls in the second quadrant, where sine is positive and cosine is negative, resulting in a negative tangent value.