Problem 18 - Technical Mathematics with Calculus

Question

Convert to degrees (decimal). $$0.116 \mathrm{rad}$$

Step-by-Step Solution

Answer

Approximately 6.648 degrees.

Step 1: Understanding Radian to Degree Conversion

The relationship between radians and degrees is that one full rotation around a circle is equal to 2\(\pi\) radians or 360 degrees. This means that \(1\) radian is equal to \(\frac{360^\circ}{2\pi}\) degrees. Therefore, to convert radians to degrees, the given radian value needs to be multiplied by this fraction.

Step 2: Applying the Conversion Factor

Use the conversion formula which states that degrees = radians \(\times\) \(\frac{180^\circ}{\pi}\). Multiply the given radian value of \(0.116\) by the conversion factor \(\frac{180^\circ}{\pi}\).

Step 3: Perform the Calculation

Calculate \(0.116 \times \frac{180^\circ}{\pi}\), which gives the value in degrees. Remember to use the value of \(\pi\) to an appropriate level of precision, for most practical purposes, \(\pi\approx3.14159\) is sufficient.

Key Concepts

Radians

Radians are a fundamental unit of measure for angles used primarily in the field of trigonometry. Unlike degrees, a radian measures an angle based on the radius of the circle. Specifically, one radian corresponds to the angle formed at the center of a circle by an arc whose length is equal to the radius of the circle.

This means that in a full circle with a circumference of \(2\pi r\), where \(r\) is the radius, there are \(2\pi\) radians. Consequently, a straight angle, which is half of a full rotation and known as \(\pi\) radians, is equivalent to 180°. When solving problems involving angles in trigonometry, the use of radians often simplifies the calculations, as many trigonometric functions take radian measures as their arguments.

Degrees

Degrees are a unit of measurement for angles commonly used in various fields such as geometry, navigation, and astronomy. One degree, symbolized as \(1^\circ\), represents \(\frac{1}{360}\) of a full rotation. Therefore, a full circle is 360°. This division of a circle into 360 parts comes from ancient civilizations, which found that using a base-60 system allowed for easier fraction handling and subdivision of the circle.

The degree system is widely used in daily life and in disciplines that require practical measurements of angles, such as in surveying and construction. Understanding how to work with degrees is essential when analyzing angles visualized in a circular plane, and it is a fundamental concept in trigonometry.

Unit Conversion

Unit conversion between radians and degrees is crucial in various scientific and mathematical contexts, as it allows for a seamless transition between the two primary measurements of angles. The basic principle behind unit conversion is using a conversion factor that derives from the proportional relationship between the units.

In this case, we know that \(2\pi\) radians is equivalent to 360°. To convert radians to degrees, you multiply by the conversion factor \(\frac{180^\circ}{\pi}\), conversely, to convert degrees to radians, you multiply by \(\frac{\pi}{180^\circ}\). Remember, when performing these conversions, maintaining the precision of \(\pi\) is important for an accurate result, and using standardized conversion factors ensures consistency across different problems and applications.

Trigonometry

Trigonometry is a branch of mathematics that focuses on the study of triangles, specifically right-angled triangles, and the relationships between their angles and sides. The field extends to cover circular functions and waves, making the understanding of radians and degrees fundamental to the study.

Trigonometric functions such as sine, cosine, and tangent are defined using the angles of the triangles, and they have properties that are most naturally described in terms of radians. For instance, in calculus and most real-world applications, radians are the default unit for measuring angles because the trigonometric functions have simpler and more universal derivatives when their inputs are in radians. Furthermore, many physical phenomena, like the oscillations of a pendulum, are described by equations that naturally involve trigonometric functions and radians.