High School: Functions
Trigonometric Functions HSF-TF.A.1
1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Although students' brains are wired to think in degrees, math theory is designed to work in radians and there's no way around it. They should understand that radians are based on the radius of a circle, and that one radian is approximately 57.3°.
Why use such a weird, random number? First of all, it isn't random. If we draw a circle with radius r, and then mark off an arc on our circle with length r, the central angle that subtends our arc is equal to 1 radian.
Students might benefit from snipping a string of length r to use to mark off the correct arc length on the unit circle. That way, they'll actually understand the definition instead of simply memorizing it. Hopefully.
A circle has 360° or 2π radians. A semicircle has 180° or π radians. That means students can multiply by or to convert from degrees to radians or vice versa.
Students should use this conversion factor properly to find both degree and radian measures of angles. Remind them that the units they're looking for (either degrees or radians) should be on the top of the conversion factor. The droids they're looking for, however…
Radians can be intimidating to students. Their whole understanding of angles and circles is being uprooted and replaced with something completely different. Giving lot of simple sketches comparing common angles in degrees with their measures in radians may be helpful.