Everyone, welcome back. As we've been discussing angles, we've gotten used to the concept of degrees. If you go around a circle, you complete 360 degrees. If I ask you to illustrate a 120-degree angle, you'd be able to draw something resembling this. In this video, we explore another unit for measuring angles called radians. Similar to using inches and centimeters to measure distance, we can use degrees and radians to measure angles.

We'll be using radians extensively in later videos when discussing functions and graphing, but for now, I just want you to understand the basic difference between degrees and radians, how to visualize them, and importantly, how to convert between them. So, let's get started. A radian is another unit for measuring angles based directly on the circumference of a circle. Observe that if you travel a full circle, which amounts to 360 degrees, you have essentially spanned 360 tiny divisions, each representing one degree. A radian, meanwhile, is derived from encircling the radius around the circumference. If you stretch the circle's radius along its circumference, the spanned arc corresponds to an angle of 1 radian, roughly 57 degrees.

Completing the circle with this measurement, you reach approximately 2π radians. Therefore, a full circle is 360 degrees or 2π radians. We often fractionalize radians in terms of π, just as we quarter 360 degrees. Halfway around the circle, you achieve π radians, and a quarter circle corresponds to π2 radians. Three quarters around is 3π2 radians. Converting between degrees and radians primarily involves the formulas π180 or 180π.

Let's explore conversion through an example. To convert 120 degrees to radians, we apply π180. Simplifying 120180, we get 23 or 2π3 radians. Conversely, to convert 6π5 radians to degrees, we apply the 180π. After cancellation and calculation, we find it to be 216 degrees. Thank you for watching, and I’ll see you in the next video.