Question

Convert the following angles in radians to degrees: (a) \(\frac{\pi}{2}\) (b) \(\frac{\pi}{3}\) (c) \(\frac{4 \pi}{3}\) (d) \(1.25 \pi\) (e) \(1.25\) (f) \(9.6314(\mathrm{~g}) 3\)

Short answer

Question: Convert the following angles from radians to degrees: (a) \(\frac{\pi}{2}\) radians (b) \(\frac{\pi}{3}\) radians (c) \(\frac{4 \pi}{3}\) radians (d) \(1.25\pi\) radians (e) 1.25 radians (f) 9.6314 radians (g) 3 radians Answer: (a) 90° (b) 60° (c) 240° (d) 225° (e) 71.62° (f) 551.85° (g) 171.89°

Step-by-step solution

Converting radians to degrees using the conversion factor

Multiply each angle in radians by the conversion factor \(\frac{180}{\pi}\) to obtain its equivalent angle in degrees. (a) \(\frac{\pi}{2} \cdot \frac{180}{\pi}\) = 90° (b) \(\frac{\pi}{3} \cdot \frac{180}{\pi}\) = 60° (c) \(\frac{4 \pi}{3} \cdot \frac{180}{\pi}\) = 240° (d) \(1.25\pi \cdot \frac{180}{\pi} \)= \(225\)° (e) \(1.25 \cdot \frac{180}{\pi}\) = \(71.62\)° (rounded to 2 decimal places) (f) \(9.6314 \cdot \frac{180}{\pi} = 551.85\)° (rounded to 2 decimal places) (g) \(3 \cdot \frac{180}{\pi} = 171.89\)° (rounded to 2 decimal places) The angles in degrees are: (a) 90° (b) 60° (c) 240° (d) 225° (e) 71.62° (f) 551.85° (g) 171.89°

Key concepts

Radian to Degree Conversion

Radian to degree conversion is a crucial skill in mathematics, particularly in trigonometry and geometry. To convert an angle from radians to degrees, one must understand the relationship between these two units. A full circle is 360 degrees but is also equal to \(2\pi\) radians. This means that \(1\pi\) radian is equivalent to 180 degrees.The conversion formula is simple: multiply the number of radians by \(\frac{180}{\pi}\). This formula comes from the equivalence \(\pi\) radians = 180 degrees. For example:

• For \(\frac{\pi}{2}\) radians: \(\frac{\pi}{2} \times \frac{180}{\pi} = 90\) degrees.
• For \(1.25\pi\) radians: \(1.25 \cdot \frac{180}{\pi} = 225\) degrees.

By applying this conversion process, any angle given in radians can be accurately changed to degrees, facilitating easier understanding and application in various mathematical problems. This conversion is particularly helpful in contexts where degrees are more commonly used, such as navigation or engineering.

Mathematical Conversion Factors

Mathematical conversion factors are constants used to translate one unit of measurement into another. In the context of angles, the conversion factor for radians to degrees is \(\frac{180}{\pi}\). This factor is derived from the fundamental relationship of the units where \(\pi\) radians equals 180 degrees.Using conversion factors efficiently requires setting up the problem correctly. Always ensure the units in question cancel correctly:

• If starting with radians, multiply by \(\frac{180}{\pi}\) to get degrees.
• If starting with degrees, you can convert back to radians by multiplying by \(\frac{\pi}{180}\).

Conversion factors simplify the process by providing a clear, consistent method to change units, making calculations faster and reducing errors. When used correctly, they are powerful tools in a mathematician's toolkit, enabling seamless transitions between different scales and units without confusion.

Angle Measurement

Angle measurement is an essential concept in both geometry and trigonometry, as angles define the shape, direction, and rotation in figures and coordinates. There are two primary systems for measuring angles: degrees and radians. Degrees are the more familiar measurement system, especially in everyday language, representing a circle divided into 360 equal parts. This system is intuitive and easy to visualize in geometric contexts. Radians, however, provide a more natural measure growing from the properties of circles. A radian is based on the radius length of a circle, making it a more intrinsic measure in many mathematical computations and analyses. When solving real-world problems or engaging in academic mathematics, understanding both systems and their conversions is vital:

• Degrees are common in fields like carpentry, navigation, and sports.
• Radians are indispensable in higher mathematics, engineering, and physics.

Thus, whether working in pure mathematics or applied sciences, a complete understanding of angle measurement and conversion is foundational for success.