Problem 81 - A Graphical Approach to Precalculus with Limits

Question

Convert each degree measure to radians. Round to the nearest hundredth. $$64.29^{\circ}$$

Step-by-Step Solution

Answer

64.29° is approximately 1.12 radians.

Step 1: Understand the Conversion Factor

To convert degrees to radians, use the conversion factor \( \frac{\pi}{180} \). This is because one complete revolution is \(360\) degrees in a circle and \(2\pi\) radians.

Step 2: Write the Conversion Formula

The formula to convert degrees to radians is \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \).

Step 3: Substitute the Degree Measure

Substitute the given degree measure, \(64.29^{\circ}\), into the formula. This gives us \( 64.29 \times \frac{\pi}{180} \).

Step 4: Calculate the Radian Measure

Multiply \(64.29\) by \(\frac{\pi}{180}\). To simplify, first compute \( \frac{\pi}{180} \approx 0.0174533 \). Then multiply: \( 64.29 \times 0.0174533 \approx 1.12166 \).

Step 5: Round the Result

Round the result from Step 4 to the nearest hundredth. Thus, \(1.12166\) rounds to \(1.12\).

Key Concepts

Radians

Radians are a way to measure angles. Unlike degrees, which divide a full circle into 360 parts, radians are based on the radius of a circle. When using radians, a full circle is equal to \(2\pi\) radians. This means that one radian equals the angle formed when the arc length of the circle is equal to the radius.
Some advantages of using radians in mathematical formulas are that it simplifies calculations in calculus and trigonometry. For example, the derivative and integral formulas for trigonometric functions are simpler when angles are measured in radians. Moreover, radians are widely used in physics, particularly in concepts involving rotational motion. Radians provide a more natural and direct way to describe angles compared to degrees.

Degrees

Degrees are the most common unit used to measure angles in everyday contexts. A full circle is divided into 360 equal parts, meaning one complete revolution equals 360 degrees (\(360^{\circ}\)). This division is based on the ancient Babylonian system, which uses a base of 60.
Degrees are simple to understand and commonly used in various practical fields, such as navigation, astronomy, and geography. They allow for clear and straightforward communication of angles, especially in educational and instructional settings. While practical, degrees can be less convenient than radians for mathematical calculations, particularly in higher-level mathematics where radian measure is preferred.

Conversion Factor

Converting from degrees to radians, or vice versa, involves using a conversion factor. The most important conversion factor to remember is that:\[1^{\circ} = \frac{\pi}{180} \, ext{radians}\] This conversion factor comes from the relation that a full circle is \(360\) degrees and also \(2\pi\) radians. Therefore, to convert degrees into radians, multiply the number of degrees by \(\frac{\pi}{180}\).
For example, to convert \(64.29^{\circ}\) to radians, use the conversion factor by calculating \(64.29 \times \frac{\pi}{180}\). This makes converting between the two units straightforward once the conversion factor is understood.

Mathematical Formulas

Mathematical formulas are essential tools for converting between different units of measurement like degrees and radians. The conversion from degrees to radians relies on the formula:
\[\text{Radians} = \text{Degrees} \times \frac{\pi}{180} \]This formula ensures accuracy and consistency in converting angles.
In practical application, when you have an angle in degrees, simply multiply it by \(\frac{\pi}{180}\) to get the equivalent angle in radians. Similarly, to convert from radians back to degrees, you would use:
\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]Understanding and utilizing these formulas effectively allows one to seamlessly work between the two angle measurements, benefiting tasks in geometry, trigonometry, and beyond.