Visualizing the Unit Circle

So you’re in math class and you’ve got all these weird values staring at you from the unit circle such as \frac{7\pi}{6} and you’re not really sure what all that means or how in the world you’ll remember what is where. Never fear! Here’s an easy method that only requires basic knowledge of fractions to understand.

Unit circle diagram from wikipedia

Now I know this all looks pretty scary and intimidating, but if you look at the angle measures in radians, and the fractions attached to the \pi in each case, you’ll see it’s not so hard to figure out after all.

We know that

  • 0 radians = 0 degrees
  • \frac{\pi}{2} radians = 90 degrees
  • \pi radians = 180 degrees
  • \frac{3\pi}{2} radians = 270 degrees
  • 2\pi radians = 360 degrees

These will be our starting points. As long as you know where 0, 0.5, 1, 1.5, and 2 are on a number line you should be fine.

Take for an example some random radian measure like \frac{5\pi}{3}. How would you figure out where it was on the unit circle without looking at the diagram? Don’t let the \pi scare you off. Let’s just look at the fraction separately.

\frac{5}{3} is almost 2 (aka \frac{6}{3}), but not quite. However, it is larger than 1.5 (that would have been \frac{4.5}{3}). So from this, we can tell that \frac{5\pi}{3} is in quadrant 4.

That wasn’t so bad, was it? As long as you know where the fraction lies between our 4 “base points”, its easy to figure out the general location of an angle on the unit circle.

Hope this helps someone, let me know if you have questions.


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