Table of Contents

Radians and Degrees

Definitions

Standard Position

An angle is in standard position if the initial side of the angle is on the x-axis between the 1st and 4th quarter.

Positive Angle

An angle that extends counterclockwise1).

Negative Angle

An angle that extends clockwise.

Coterminal Angle

Equivalent angle that has a different numerical value. Add or subtract $360$ for degrees and $2\pi$ for radians to get a coterminal angle.

Degrees

Degrees is an angle measurement where one full revolution is 360.

Degrees Minutes Seconds

Instead of a decimal representation, degrees can also be specified in minutes and seconds, akin to standard time2). The conversion to decimal is $x + \frac{y}{60} + \frac{z}{3600}$ where x is whole unit degrees, y is minutes3), and z is seconds4).

Radians

A radian is a measurable length where the radius of a circle equals to an arc length section of the circle. $r = s$ where $r$ is the radius and $s$ is the arc length segment.

Converting Rad to Deg and vice versa

The conversion is $\pi = 180$ degrees. Degree to Radian: $d * \frac{\pi}{180} = r$ Radian to Degree: $r * \frac{180}{\pi} = d$

Complementary and Supplementary Angles

1)
a common misconception is that angles are positive when rotated clockwise, the standard in mathematics is the opposite
2)
which supposedly degrees are based on
3)
denoted by ' mark/foot mark
4)
denoted by “ mark/inches mark