Problem 49 - Precalculus: A Concise Course

Question

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$ \sin \frac{\pi}{4} $$

Step-by-Step Solution

Answer

The trigonometric function \(\sin \frac{\pi}{4}\) is approximately 0.7071 when rounded to four decimal places.

Step 1: Set calculator to radian mode

Ensure the calculator is in radian mode. This is because the angle we're dealing with, \(\frac{\pi}{4}\), is given in radians.

Step 2: Evaluate Sin(\(\frac{\pi}{4}\))

Input into the calculator the following: sin(\(\frac{\pi}{4}\)). Press the equals sign on the calculator to obtain the result.

Step 3: Round the result

The result will be given as a decimal. Round this decimal to four decimal places as directed by the problem statement.

Key Concepts

Radian Mode Calculator

When dealing with trigonometric functions, ensuring that your calculator is in the correct mode is crucial for getting accurate results. For angles measured in radians, a calculator must be switched to radian mode. To do this, look for a mode button on your calculator, which often allows you to cycle between degrees, radians, and sometimes gradians. Select radians as your preferred unit of angle measurement.

For exercises involving π (pi), such as evaluating \( \sin\frac{\pi}{4} \), it's important because π radians is equivalent to 180 degrees. Therefore, to input angles in terms of π correctly, the radian mode is necessary. Always remember to switch back to degree mode if your next calculations involve angles in degrees to avoid confusion and incorrect results.

Sine Function

The sine function is a fundamental trigonometric function that gives the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle. This function is periodic and oscillates between -1 and 1. In the unit circle, where the radius is 1, the sine of an angle is the y-coordinate of the corresponding point on the circle.

For example, evaluating \( \sin\frac{\pi}{4} \) involves finding the y-coordinate of the point on the unit circle that is \(\frac{\pi}{4}\) radians away from the rightward horizontal axis. Since \( \frac{\pi}{4} \) is 45 degrees, and the unit circle is symmetrical, the sine of 45 degrees (or \(\frac{\pi}{4}\) radians) is \(\frac{\sqrt{2}}{2}\), which is approximately 0.7071. Understanding these relationships and how they correspond to the unit circle is essential for mastering the sine function.

Rounding Decimals

Rounding decimals is a mathematical process used to shorten a number to a desired degree of precision. This is often needed when dealing with irrational numbers or when a certain level of precision is sufficient for practical purposes. For instance, in trigonometry, it's common to round answers to several decimal places.

To round a decimal to four places, you look at the fifth digit. If this digit is 5 or more, you increase the fourth digit by one. If it is less than 5, the fourth digit stays as is and all subsequent digits are dropped. In our example, if the calculator shows an answer of 0.70710678 for \( \sin\frac{\pi}{4} \), we round it to 0.7071 because the fifth digit (0) is less than 5. Remember, rounding can affect the accuracy of your results, so always use the unrounded number for subsequent calculations if necessary.