Problem 72 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the angle measure from degrees to radians. Round your answer to three decimal places. $$83.7^{\circ}$$

Step-by-Step Solution

Answer

The radian measure of the angle \(83.7^\circ\) rounded off to three decimal places is 1.461 radians.

Step 1: Identify the given degree measurement

We are given one angle measurement to convert from degrees to radians. The angle measurement is \(83.7^\circ\).

Step 2: Convert degrees to radians

To convert the degree measurement into radians, we use the conversion factor \( \frac{\pi}{180} \). Therefore, we multiply \(83.7^\circ\) by \( \frac{\pi}{180} \) to convert it to radians. So, \(83.7^\circ \times \frac{\pi}{180} \).

Step 3: Calculate and Round off to three decimal places

We perform the multiplication and get the answer in radians. We then round off our answer to three decimal places. If the fourth decimal digit is less than 5 we ignore it. If it's 5 or more, we increase the third decimal digit by 1.

Key Concepts

Angle Measurement

Angles can be measured in several ways, with the most common units being degrees and radians. Understanding these units is crucial in various fields, including mathematics, physics, and engineering.

• Degrees: A full circle is 360 degrees, meaning one degree is \(\frac{1}{360}\) of a circle.
• Radians: A full circle is \(2\pi\) radians, where \(\pi\approx3.14159\), so one radian is \(\frac{180}{\pi}\) degrees.

Converting between these units allows for precise calculations regarding angles, be they for trigonometric functions or geometrical figures. Understanding how to work with angles in both degrees and radians is essential for mathematicians and scientists to communicate measurements accurately.

Degree to Radian Conversion

Converting an angle from degrees to radians is straightforward when you use the conversion factor \(\frac{\pi}{180}\). This factor comes from the relationship between the total degrees and the radians of a circle:

• 180 degrees equals \(\pi\) radians.
• To convert degrees to radians, multiply the number of degrees by \(\frac{\pi}{180}\).

For example, when converting 83.7 degrees, you calculate \(83.7 \times \frac{\pi}{180}\) to find its equivalent value in radians. This action translates the angle's measure into the unit that fits different mathematical applications, especially in calculus and trigonometry.

Rounding Decimals

Once you have converted an angle to radians, you often need to express the result to a specific number of decimal places. In mathematics, rounding helps make results more practical and easier to read, especially in scientific calculations.

To round a number to three decimal places, follow these steps:

• Look at the fourth decimal digit.
• If it is less than 5, round down by leaving the third decimal digit as is.
• If it is 5 or greater, round up by increasing the third decimal digit by 1.

By following these rounding rules, you ensure your answer is precise to the desired degree of accuracy, which is particularly useful when presenting answers in both academic and real-world applications.