The above Chart(s) are two diagrams to express the unit circle in Degrees or Radians.
For some of us, counting with Pi may be second nature. However, I know how frustrating not understanding something can be. So, instead of explaining the formulas to convert degrees to radians and to convert radians back to degrees, I would like to show you a cool trick how to count with Pi without knowing the conversion formulas!
First, let’s establish the most important angle needed to count with Pi:
30° – Aka: π/6
As you can see from the two charts above, 30° and π/6 are equal to one another. π/6 is the way to express 30° in radians. A Radian is the type of unit used when working with Pi or circles!
OK, so let’s not forget that 30° is equivelent to π/6 in radians!!!
Also, for the sake of this lesson, we are going to skip over the following angles: 45°, 135°, 225° & 315°
The reason for this is that those four angles cannot be divided by 30. So therefore, they cannot fall under the rule/trick I am about to show you. However, all four angles are divisible by 45°. So once you have a good understanding of how this trick works with 30°, you should be able to apply the same method for the four angles divisible by 45°. Interesting enough though, if you are learning how to work with the unit circle, already are familiar with the unit circle, or you recall the 45° – 45° triangle from Geometry, you will certainly have 45° memorized in radians very quickly. 45° is eqivelent to π/4 in radians, for the record.
Now let’s get started!
So we have established that 30° = π/6
To find the next angle in the unit circle we add the following:
π/6 + π/6 = 2π/6 = π/3
(this is 30° + 30° = 60°, therefore 60° ≈ π/3)
so now that we know what 60° is in radians (π/3), we can use it plus π/6 (remember 30°) to find our next angle in the unit circle.
depending upon how well you do or don’t know how to add fractions, may depend here how you choose to find the next angles. Let me clarify what I mean. 60° was π/3 because we actually had to simplify our answer of 2π/6. If we choose to use π/3 for our 60°, we will have to multiply the denominator of π/3 before we can add π/6 to it. Or we can take the shorter route and choose to work with the un-simplified answers so that we do not have to find the common denominator. For the sake of the work, I am going to add the angles with the un-simplified answers. i think this will help you understand better how this trick works and why certain angles are represented in π the way they are. It will also point out exactly what I am getting at here as well.
Ok, getting back to where we were….
2π/6 + π/6 = 3 π/6 = π/2 = 90°
(this is 60° + 30° = 90°, therefore 90° ≈ π/2 or 3 π/6 for our unsimplified version)
next we take 3 π/6 (90°) and add π/6 (30°) to it.
3 π/6 + π/6 = 4 π/6 = 2 π/3 = 120°
(this is 90° + 30° = 120°, therefore 120° ≈ 2 π/3 or 4 π/6 for our unsimplified version)
next we take 4 π/6 (120°) and add π/6 (30°) to it.
4 π/6 + π/6 = 5 π/6 = 150°
(this is 120° + 30° = 150°, therefore 150° ≈ 5 π/6 and lucky for us it cannot be simplified.) So we can continue.
5 π/6 + π/6 = 6 π/6 = π = 180°
(this is 150° + 30° = 180°, therefore 180° ≈ π or 6 π/6 for our unsimplified version)
…. This is where this trick really comes in handy. For the most part, when working in radians you will have some of the key angles memorized. Once you get past 180°, things kind of get confusing. One secret though from here, is that 180° actually is equivelent to 6 π/6! Remember that will help you in finding the next angles easily with a common denominator. Moving on.
6 π/6 + π/6 = 7 π/6 = 210°
(this is 180° + 30° = 210°, therefore 210° ≈ 7 π/6 and lucky for us it cannot be simplified.) So again, we can continue. You hanging in?
7 π/6 + π/6 = 8 π/6 = 4 π/3 = 240°
(this is 210° + 30° = 240°, therefore 240° ≈ 4 π/3 or 8 π/6 for our unsimplified version)
8 π/6 + π/6 = 9 π/6 = 3 π/2 = 270°
(this is 240° + 30° = 270°, therefore 270° ≈ 3 π/2 or 9 π/6 for our unsimplified version) Getting the hang of it yet?
9 π/6 + π/6 = 10 π/6 = 5 π/3 = 310°
(this is 270° + 30° = 310°, therefore 310° ≈ 5 π/3 or 10 π/6 for our unsimplified version) We are almost done! ;”]
10 π/6 + π/6 = 11 π/6 = 330°
(this is 310° + 30° = 330°, therefore 330° ≈ 11 π/6 and lucky for us it cannot be simplified.) Last one for 1 revolution and for this lesson!
11 π/6 + π/6 = 12 π/6 = 2 π = 360°
(this is 330° + 30° = 360° = 1 full revolution around the unit circle, therefore 360° ≈ 2 π or 12 π/6 for an unsimplified version)
This completes my little trick on how to count with Pi. However, you may keep going if you like and complete another revolution. Or you can apply this method for 45° like a mentioned earlier.
Hope this helped your understanding of how to count with Pi! (: