When light emits from a point, or a source like the sun, we call it a ray. In math, the space, or the amount of rotation, between any two rays from the same point creates an angle. Radian measurements relate angles back to their position in the unit circle.
Circumference, or the distance around a circle, is 2 π r, where r stands for the radius. Because circumference means all the way around the circle, or 360 degrees around, you can say that 360 degrees = 2 π radians, where a radian is dimensionless comparison to any given radius of a circle.
Because 360º = 2 π radians, you can divide both sides of the equation by 2 and get 180º = π rad. Dividing by 180º, you get 1 = π rad / 180º. Therefore, you can convert 60º to a radian measurement by multiplying by the conversion ratio, π rad / 180º. That is, 60º x π rad / 180º = π rad / 3, or π/3 radians is the equivalent of 60º.
Unlike angles in degrees, which grow much larger with rotation, mathematicians need to discuss radians in terms of reasonable proportions. For example, eight rotations of an angle is 8(360º) = 2,880º or 8(2 π rad) = 16 π radians. It is considered easier to write 16 π than 2,880º, or a number significantly larger.
Because after 2 π radians, or 360º, you return to a related angle less than 360º, radians help to figure out that "reference angle" faster. For instance, when it comes to cos(angle) in trigonometry, cos(2520º) takes longer to examine than cos(14 π rad), especially when you know that cosine of any even-integer radian amount is 1. You would have to first figure out how to make 2520º less than 360º, its reference angle, before assessing it that way. Radian measurements, therefore, simplify the burden of addressing larger angles.