Question

Use a calculator to find the approximate value of each expression. Round the answer to two decimal places. $$ \cos \frac{\pi}{8} $$

Short answer

The approximate value of \( \text{cos} \frac{\text{π}}{8} \) is 0.92.

Step-by-step solution

Understand the Expression

The given expression is \(\text{cos} \frac{\text{π}}{8}\). This means we need to find the cosine of the angle \(\frac{\text{π}}{8} \) radians.

Convert Radians to Degrees (Optional)

Since calculators usually have a mode for both degrees and radians, it is not necessary to convert. However, for understanding, \( \frac{\text{π}}{8} \) radians is equal to \(\frac{180°}{8} = 22.5° \).

Set Calculator to Radians Mode

Ensure the calculator is set to radians mode because the angle is in radians.

Enter the Expression

Enter \( \frac{\text{π}}{8}\) into the calculator in radians mode. Then use the cosine function to find the result: \( \text{cos} \frac{\text{π}}{8} \).

Read and Round the Result

The calculator should display a value. Round this value to two decimal places to get the final answer.

Key concepts

Trigonometric Functions

Trigonometric functions, like the cosine function, are fundamental tools in mathematics. They relate angles to the lengths of sides in right-angled triangles. The cosine function, specifically, gives you the ratio of the adjacent side to the hypotenuse in a right triangle. When dealing with angles, especially beyond simple geometric problems, the cosine function helps solve various applications in physics, engineering, and computer science. For example, calculating the cosine of an angle is crucial in wave motion, signal processing, and analyzing periodic phenomena.
To find the cosine of an angle, you typically use either radians or degrees. In this exercise, we are dealing with radians, which are often used in higher mathematics because they simplify many formulas. The cosine function is periodic, meaning it repeats its values in a regular interval, specifically every \(2π\) radians or \(360°\).

Radians to Degrees

Understanding how to convert between radians and degrees is important for solving trigonometric problems. Radians and degrees are two units for measuring angles. One full rotation around a circle is \(360°\), which is equivalent to \(2π\) radians. The given expression in the exercise, \(\frac{π}{8}\) radians, can be converted to degrees for better intuition. Conversion from radians to degrees uses the conversion factor \(180°/π\). So, \(\frac{π}{8}\) radians is equivalent to \(\frac{180°}{8} = 22.5°\).

This conversion is particularly useful because some students may find degrees more intuitive to understand. Calculators typically allow you to work in either unit, but knowing how to switch between them aids in comprehending the problem more deeply.

Calculator Usage

Using a calculator effectively is key to solving trigonometric problems accurately. Modern calculators have a trigonometric function mode that can handle both radians and degrees. In this exercise, since the angle is given in radians, ensure that your calculator is set to radians mode. This is often just a setting that needs to be adjusted before entering the expression.

Here are the steps to follow on your calculator:

• Make sure your calculator is in radians mode. This is usually a mode setting option called 'RAD' in the mode menu.
• Enter the value of \(\frac{π}{8}\), which you can do by typing 'π' divided by '8'.
• Use the cosine function, usually labeled 'cos', and input the previously calculated radians value.
• Read the result provided by the calculator. In this case, you should get a value that approximates to a certain number.
• Round this result to two decimal places for the final answer.

This systematic approach ensures that you get an accurate and precise result, which is essential for understanding and validating trigonometric computations.