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Angles are a way of describing the rotation or the space between two intersecting lines or surfaces. They can be measured in different units, primarily degrees and radians.
Degrees are more familiar to most people. A full circle is 360 degrees. This system is very convenient in many practical situations, like navigation or everyday measurements.
Radians, however, are an alternative way to measure angles that is especially useful in mathematics and the sciences. One radian is the angle created when the length of the arc of the circle is equal to the radius of the circle. A full circle is equal to \( 2 \pi \) radians or roughly 6.2832 radians.

The conversion between radians and degrees is crucial in trigonometry, which deals with the relationships between angles and lengths. To convert between these units, you use the fact that the angle measure of a full circle is 360 degrees or \( 2 \pi \) radians.
Since \( 2 \pi \) radians = 360 degrees, we can derive that \( \pi \) radians = 180 degrees. This is the foundation for the conversion factor used in steps:
\( \pi = 180^{\circ} \). Thus, \( 1 \) radian equals \( \frac{180}{\pi} \) degrees. This relationship allows us to easily convert from radians to degrees and vice versa by multiplication or division.

Understanding the relationship between degrees and radians is essential for solving problems in trigonometry and calculus. Knowing which system to use depends on the context; in some cases, radians are more convenient, especially in integration and differentiation in calculus.
To convert radians to degrees, you multiply the radian measure by \( \frac{180}{\pi} \). For example, for the angle \( -\frac{7 \pi}{20} \), the conversion is done as follows:
Step 1: Multiply the angle by the conversion factor:
\( -\frac{7 \pi}{20} \times \: \frac{180}{\pi} \)
Step 2: Simplify the expression by canceling out the \( \pi \):
\( -\frac{7 \times 180}{20} \)
Step 3: Further simplification leads to:
\( -63^{\circ} \)
This demonstrates how easily you can switch between the two measurement systems.