# Angles

The most popular unit for measuring angles is the degree. There is 360 degrees to a full circle, 90 degrees is the right angle. There are essentially two ways of specifying the angle between two rays.

1. If the rays are of the same importance, we get an angle that is not oriented (for instance between the sides of a triangle). Given two rays, we always take the smaller angle, measure it and express the answer as a positive number (or zero).

2. If one ray is in some way special (for instance the x-axis in a coordinate system), we usually use an oriented angle, taken positive in the counterclockwise direction (towards the y-axis) and negative in the clockwise direction.

We see that now the same angle can be specified in several ways. In fact, it can be specified in infinitely many ways, since in this situation we can go around the origin several times, each time gaining a full circle (360 degrees), and we can do it both ways.

Degrees are "practical". People use them when working with geometrical objects, calculating distances and angles "in the real world" and situations like this. Since they correspond to directions on a compass, we are pretty familiar with them. In such settings we also often use the non-oriented angle.

However, in sciences, most notably in math and physics, we (almost) always use the oriented angle and prefer a different unit for it: the radian.

Given an angle, we obtain its size in radians by dividing the arc length by the radius.

This is one great advantage of the radian, since in situations as in the picture we calculate the arc length as (angle)⋅(radius). This is very useful in theoretical calculations.

The full angle is (circumference)/(radius), that is, the full angle is 2π. Similarly we easily calculate (or guess) the important four angles:

The relationship between degrees and radians is linear, so we have simple transformation formulas:

Probably the most popular angles are these:

Why are they popular? π/2 is the right angle, its importance is hopefully clear. The angles π/4 and π/6 represent the half and the third of the right angle, they appear quite often. Finally, π/3 is exactly the angle that appears in an equilateral triangle.

Note that if we make a right-angle triangle with another angle π/3, then the third angle must be π/6. Also, an isosceles right-angle triangle has the two other angles equal to π/4. So it would seem that these angles appear in geometry rather often.

By adding/subtracting right angles we get the whole "compass rose":