Radians and degrees are two ways we measure angles. They are closely related when we look at the unit circle.

The unit circle is just a circle with a radius of 1, and it's centered at the origin of the coordinate plane.

How to Convert

1. From Degrees to Radians: To change degrees into radians, we use this formula: $\text{radians} = \frac{\pi}{180} \times \text{degrees}$ For example, to find out what a $90^\circ$ angle is in radians: $\frac{\pi}{180} \times 90 = \frac{\pi}{2} \text{ radians}$

2. From Radians to Degrees: To go from radians back to degrees, we use this formula: $\text{degrees} = \frac{180}{\pi} \times \text{radians}$ For instance, if we have an angle of $\frac{\pi}{3}$ radians, we can convert it to degrees like this: $\frac{180}{\pi} \times \frac{\pi}{3} = 60^\circ$

The Unit Circle and Angles

On the unit circle, every angle matches up with coordinates:

• At $0$ radians (or $0^\circ$), the coordinates are $(1, 0)$.
• As you move around the circle, every $90^\circ$ (or $\frac{\pi}{2}$ radians) changes where you are:
  • $0^\circ$ or $0$: $(1, 0)$
  • $90^\circ$ or $\frac{\pi}{2}$: $(0, 1)$
  • $180^\circ$ or $\pi$: $(-1, 0)$
  • $270^\circ$ or $\frac{3\pi}{2}$: $(0, -1)$

Learning about how these angles interact helps you understand trigonometric functions. You can see how they fit right into the shape of the unit circle!