Problem 34 - Precalculus

Question

Use a calculator to evaluate the trigonometric functions for the indicated angle values. Round your answers to four decimal places. $$\cos 21.9^{\circ}$$

Step-by-Step Solution

Answer

The value of \( cos 21.9^{\circ} \) is approximately 0.9281.

Step 1: Set Calculator to Degree Mode

Ensure your calculator is in degree mode, as we are dealing with an angle in degrees. This is typically done by pressing the 'Mode' button and selecting 'Degree' instead of 'Radian'.

Step 2: Input the Angle Value

Enter the angle value, 21.9, into your calculator. Make sure to input it correctly to avoid errors.

Step 3: Calculate the Cosine

Use the cosine function by pressing the 'cos' button followed by the angle you've entered. The calculator should display the cosine of 21.9 degrees.

Step 4: Round the Result

Take the result shown on your calculator and round it to four decimal places to obtain the final answer.

Key Concepts

Angle Measurement

Understanding angle measurement is crucial when working with trigonometric functions. Angles are typically measured in degrees or radians, depending on the context. In the problem at hand, the angle is given in degrees, which is one of the most common units used in mathematics.

• A full circle is 360 degrees.
• One degree equals 1/360 of a full rotation.

For trigonometric functions like cosine, it is important to be aware of the angle's measurement unit to ensure accurate calculations. When dealing with degrees, visualize the rotation around a circle and how it relates to the angle given. This can help in predicting and understanding the behavior of trigonometric functions.

Degree Mode

When working on problems involving angles measured in degrees, it is essential to set your calculator to degree mode. Calculators can operate in either degree or radian mode, and using the wrong mode can result in incorrect answers.
To set your calculator to degree mode:

• Generally, press a button labeled 'Mode' or 'Angle'.
• Select 'Degree' from the options displayed, rather than 'Radian'.

Ensuring your calculator’s correct mode is a small but vital step. It aligns the calculator's internal settings with the angle measurement given, leading to accurate results while evaluating trigonometric functions.

Cosine Calculation

Calculating the cosine of an angle involves using the cosine function, a fundamental part of trigonometry that relates the ratio of the adjacent side to the hypotenuse in a right triangle. For this exercise:

• Enter the angle (21.9 degrees) into your calculator.
• Press the 'cos' button to obtain the cosine value.

This output represents the cosine of the angle 21.9 degrees when measured from the positive x-axis. Cosine calculations are commonly used in solving real-world problems like determining the lengths of sides in a triangle or modeling cycles and waves. Once you have the cosine value, round it to four decimal places. This rounding is often necessary in homework or test situations where increased precision is crucial for demonstrating understanding.