Problem 18 - Basic Technical Mathematics with Calculus

Question

By means of the definition of a radian, change the given angles in radians to equal angles expressed in degrees to the nearest \(0.01^{\circ}\). $$0.838 \mathrm{rad}$$

Step-by-Step Solution

Answer

\(0.838 \) radians is approximately \(47.99^{\circ}\).

Step 1: Understanding Radians and Degrees

Radians and degrees are two ways of measuring angles. A full circle in radians is \(2 \pi\) while in degrees, it's \(360^{\circ}\). We can use this relationship to convert radians into degrees.

Step 2: Writing the Conversion Formula

To convert radians to degrees, we use the formula: \( \text{Degrees} = \text{Radians} \times \left( \frac{180}{\pi} \right) \). This formula arises from the fact that \(2\pi\) rad equals \(360^{\circ}\).

Step 3: Plugging in the Value

Insert the given radian measurement into the formula. Here, \(0.838\) is the angle in radians, so it becomes: \(\text{Degrees} = 0.838 \times \left( \frac{180}{\pi} \right) \).

Step 4: Calculating the Conversion

Compute the multiplication: \(0.838 \times \left( \frac{180}{\pi} \right) \approx 0.838 \times 57.2958 = 47.99 \). This result is in degrees.

Step 5: Rounding the Result

Round the computed degree measurement to the nearest hundredth of a degree, which is already done and results in \(47.99^{\circ}\).

Key Concepts

Radian to Degree Conversion

Converting radians to degrees is a fundamental concept in trigonometry that helps in understanding angles in different measurement systems. Since radians and degrees are both units for measuring angles, knowing how to switch between them is crucial. The conversion uses the relationship between the total radians and degrees in a circle. A complete circle is measured as \( 2\pi \) radians or \( 360^{\circ} \). Therefore, the conversion formula is derived as follows:

• From radians to degrees: \( \text{Degrees} = \text{Radians} \times \left( \frac{180}{\pi} \right) \)
• From degrees to radians: \( \text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right) \)

Using this conversion is especially useful in various fields such as physics, engineering, and computer science, where different conventions of angle measurement are prevalent.

Degree Measurement

Degrees represent a traditional way to quantify angles, based on dividing a circle into \( 360 \) equal parts. One degree is \( \frac{1}{360} \) of a complete circle. This unit has its origins in ancient Babylonian astronomy, where they used a base 60 number system which made \( 360 \) a highly divisible choice.
A degree is further divided into minutes and seconds, where

• 1 degree (\(1^{\circ}\)) = 60 arcminutes (\(60'\))
• 1 arcminute (\(1'\)) = 60 arcseconds (\(60''\))

Most importantly, degrees provide a more intuitive understanding of angles for many practical applications, making them essential in fields such as navigation, mapping, and everyday geometry.

Angle Measurement

Angle measurement is critical in understanding geometry, trigonometry, physics, and various other subjects. It helps us define the size of the angle formed when two lines or surfaces intersect.
The core units of angle measurement are radians and degrees:

• Radians: Based on the radius of a circle; more scientific and mathematically pure.
• Degrees: A traditional unit, dividing a circle into 360 parts; more intuitive for everyday use.

Both units can be used interchangeably with proper conversion, providing flexibility in mathematical calculations and interpretations. Choosing between radians and degrees often depends on the context and the convention used in specific fields or disciplines.

Radians

Radians offer a unique perspective by defining angles relative to the radius of a circle. When an angle is one radian, the arc it "subtends" or "cuts off" is equal to the radius. This makes radians particularly suitable for advanced mathematics and calculus, as they simplify many formulas, such as those involving trigonometric functions and their derivatives.
Key attributes of radians include:

• More naturally align with the properties of circles and trigonometric identities.
• Simplify the complexity of equations in various branches of mathematics and physics.
• Facilitate direct applications of calculations involving angular velocity or periodic functions.

Despite seeming less intuitive than degrees, radians are preferred in many scientific and engineering calculations because they provide a natural and consistent way to measure angles related to circular motion.