Problem 37 - Precalculus

Question

Use a calculator to evaluate the trigonometric functions for the indicated angle values. Round your answers to four decimal places. $$\cos \left(\frac{6 \pi}{5}\right)$$

Step-by-Step Solution

Answer

\(\cos \left(\frac{6 \pi}{5}\right) \approx -0.8090\)

Step 1: Convert Radians to Degrees

First, we convert the angle from radians to degrees. We use the conversion factor that \(180^{\circ} = \pi\) radians. Thus, \(\frac{6\pi}{5}\) radians is equivalent to \(\frac{6\pi}{5} \cdot \frac{180}{\pi} = 216^{\circ}\).

Step 2: Use the Calculator for Cosine

Input \(216^{\circ}\) into your calculator to determine \(\cos 216^{\circ}\). Make sure the calculator is set to degree mode, not radian mode. The result is approximately -0.8090.

Key Concepts

Angle Conversion

Understanding angle conversion is crucial when working with trigonometric functions, especially when angles are given in radians but we prefer using degrees. Radians and degrees are two different ways of measuring angles.
Converting between the two involves a straightforward formula, which is based on the key equivalence: 180 degrees is equal to \( \pi \) radians. When you have an angle measured in radians, like \( \frac{6\pi}{5} \), converting it to degrees involves multiplying by \( \frac{180}{\pi} \). Here's why:

• Multiply the radian measure by \( \frac{180}{\pi} \) to get degrees.
• In this specific example: \( \frac{6\pi}{5} \cdot \frac{180}{\pi} = 216^{\circ} \).

This conversion is essential because it allows you to work with angles in degrees, a more intuitive unit for many people.

Cosine Function

The cosine function is one of the primary trigonometric functions and is denoted as \( \cos \theta \), where \( \theta \) is an angle. This function relates the angle to the ratio of the adjacent side over the hypotenuse in a right triangle.
When you need to find the cosine of an angle that is beyond 90 degrees, such as 216 degrees, it refers to evaluating the cosine function over the complete unit circle.

• For angles greater than 180 degrees but less than 270 degrees, like 216 degrees, the cosine value is negative.
• This is due to the position of the angle in the unit circle, lying in the third quadrant where cosine values are negative.

For \( \cos(216^{\circ}) \), the value is approximately -0.8090, reflecting its position in this quadrant. Understanding the unit circle is helpful for such evaluations.

Calculator Usage

Using a calculator effectively is vital for solving trigonometric problems. Ensure your calculator is set to the correct mode that matches your angle measurement unit, either degrees or radians.
Here are some tips for using a calculator to find trigonometric values:

• First, confirm the angle mode on your calculator by checking if it displays ":DEG" for degrees or ":RAD" for radians.
• For our example, switch to degree mode because we're calculating \( \cos(216^{\circ}) \).
• Directly input the angle value and select the cosine function to obtain the result.
• Round the outcome if necessary; here, \( \cos(216^{\circ}) \) results in -0.8090, rounded to four decimal places.

These practices ensure that your calculations are accurate and help you avoid common errors related to angle measurement discrepancies.