Divisible By Pi

This is an adaptation of an explanatory post I did, taken from its natural environment, slapped around a bit and put here for your viewing pleasure. Enjoy.

We’ll start with the short answer: All of the nicest angles are divisible by pi. If you’ve done any first year university maths subjects, this should make immediate sense to you — the rest of you, likely, are going to require a bit more convincing.

The first question you should be asking yourself is, “How is he measuring angles?” The common understanding of how angles are measured – i.e. in degrees – clearly makes no sense here. A circle has 360 degrees, a number that doesn’t lend itself easily to finding multiples of pi lurking around.

The units that we use to measure the angle – and that work out in the end to be much more convenient and intuitive once you’ve got a good grasp of them, are called ‘radians’ (abbreviated to ‘rads’) and are related to degrees by the simple formula . If this seems like an odd measure to use, multiply both sides of the equation to get , and remember the old high school equation for circumference of a circle: . Voila!

The next bit of high school trig to remember are the rules for the sine, cosine and tangent functions commonly abbreviated as “SOH CAH TOA”. For those of you who don’t remember, SOH CAH TOA is a mnemonic for the three equations:

Where ‘A’, ‘O’ and ‘H’ stand for ‘Adjacent’, ‘Opposite’ and ‘Hypotenuse’, as in the diagram on the left. As can be clearly seen in the diagram, adjacent and opposite refer to the position of the sides with reference to the angle theta that we are interested in, and the hypotenuse refers to the long side opposite the right angle in a right angle triangle.

(I know, I know – you remember all this…just bear with me, in case there are people who don’t.)

Now that we’ve got all that down, you might be able to see where I’m going with all this – given that the sine, cosine and tangent functions can be thought of as reporting the ratio of two sides in a right angle triangle (with caveats and tricks applied along the way that aren’t important here), the nicest angles are the ones that report back nice ratios when plugged into a trigonometric function. Aside from anything else, such angles and useful when you have to make approximations – it’s much easier to deal with a simple ratio than a value like .

All of this brings me to the final part of the post – this picture:

If you’ve followed me along this far, you can probably guess that this has really just turned into a game of ‘fill in the blanks’. For the triangle on the left, the non-right angles are both , and for the triangle on the right the bottom angle is and the top angle is . Notice anything in particular?

That’s right. They’re all divisible by .