Problem 44 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the giren trigonometric functions to four decimal places. $$\cos \frac{\pi}{5}$$

Step-by-Step Solution

Answer

The cosine of \(\frac{\pi}{5}\) calculated to four decimal places using a scientific calculator is 0.8090.

Step 1: Understand the Problem

We are required to evaluate the cosine of \(\frac{\pi}{5}\). The cosine function is a trigonometric function that can be evaluated using a scientific calculator.

Step 2: Use the Scientific Calculator

Enter \(\frac{\pi}{5}\) into the calculator and press the cosine function to calculate the result.

Step 3: Report the Answer

Ensure that the answer is reported to four decimal places as stated in the question.

Key Concepts

Scientific Calculator Use

Understanding how to use a scientific calculator effectively is essential for students, especially when dealing with trigonometric functions such as the cosine function. A scientific calculator is designed to handle not just basic arithmetic but also complex mathematical operations, including those needed in trigonometry, calculus, and physics.

When it comes to using a scientific calculator to evaluate trigonometric functions, the process is typically straightforward:

• Identify which trigonometric function you need to use, such as sine, cosine, or tangent.
• Ensure that your calculator is in the correct mode (degrees or radians) based on the given problem.
• Input the angle value into the calculator.
• Press the corresponding function key (e.g., COS for cosine).
• Read the result displayed on the screen.

Remember to check if your calculator is set to display the correct number of decimal places, and adjust the settings if necessary to obtain the precision required for your calculations.

Cosine Function

The cosine function is a fundamental concept in trigonometry and represents one of the primary trigonometric ratios. It is defined for an angle within a right-angled triangle as the ratio of the length of the adjacent side to the length of the hypotenuse.

In a unit circle setting, where the radius is 1, the cosine of an angle corresponds to the x-coordinate of a point on the circle's circumference. Hence, the cosine function relates an angle to the horizontal distance in the circular motion.

Properties of the cosine function include:

• It has a range from -1 to 1.
• It is an even function, meaning \(\cos(-\theta) = \cos(\theta)\).
• Shows periodicity with a period of \(2\pi\) radians or 360 degrees.

Understanding these properties helps in interpreting the values produced by the function when calculating angles using a scientific calculator.

Trigonometric Values Precision

Precision in trigonometric values is crucial, particularly in fields such as engineering, sciences, and mathematics where accurate measurements are necessary. When evaluating trigonometric functions, returning a precise value is essential for reliable results, and thus stating the correct number of significant figures or decimal places is a key consideration.

When using a scientific calculator to determine trigonometric values, you may encounter a default precision setting. However, our ability to adjust this setting allows us to conform to the requirements of a specific problem. For instance, the problem may require the answer to be rounded to four decimal places. Here's why precision matters:

• Ensures consistency and uniformity in calculations.
• Eliminates potential rounding errors that could affect subsequent calculations.
• Provides a level of detail that could be necessary for modeling or predicting outcomes accurately.

To achieve precision when evaluating trigonometric functions, take note of the rounding rules and be mindful of the calculators' capacity to store and display decimal places.