CONCEPT 1 – What is a radian?

Up to this point we have measured angles only in terms of degrees.  In most work involving applications of trigonometry, angles are measured in degrees.  In more advanced work in mathematics, the use of radian measure of angles is preferred.  Radian measure allows us to treat trigonometric functions and other angle-based functions as functions with domains of real numbers (distances), rather than angles. 

So what is a radian?

A radian is the central angle that subtends an arc length of one radius.

CONCEPT 2 – Converting between Degrees and Radians

One of the necessary skills is to be able to convert between degrees and radians.  These two types of angle measures have different strengths and weaknesses in certain areas and so converting in and out of them is an important skill.  We will see how some of the typical degree based formulas become greatly simplified when working in radians.
To be able to convert between two different measurements we need two values where they have the same value – one complete circular revolution.  In that case the degree measure is 360° and the radian measure is 2Π radians.  So 360° = 2Π radians.

CONCEPT 3 – Radians and Arc Length

CONCEPT 4 – Radians and Area

Much like the way we dealt with the length of an arc we do the same with the area of a sector.  Think of a sector as a part or percentage of the whole area of the circle.  We adjust the area formula slightly by creating a percentage of the whole, in the case of degrees it is a percentage of 360° and in the case of radians it is a percentage of 2Π.

CONCEPT 4 – Application Problems

Pulleys & Gears