The trigonometric circle

Short 101 on about the basis of trigonometry.

I always regretted not learning maths. My current level is, let’s say, below sea. Deep deep deep inside the ocean. I’m planning to change that, so today I’m gonna start with trigonometry . It may seem basic, but honestly that’s where I feel like math stopped making sense.

In this post I will attempt to illustrate angle measurements, angluar velocity, the basic trigonometric functions and how our favourite contant, π, fits in. And I am going to do this using the trigonometric circle.

High level concepts 🔗

What is an unit circle? 🔗

A unit circle is just a circle that has a radius with a length of 1. And just that.

The first question that I have to answer before proceeding further is why do we use this circle? After a bit of digging I ended up reading about Analytical Geometry and Calculus (a.k.a. Real Analysis). In this post I’m only going to discuss about the first one, but Calculus is on my todo list for sure.

What is Analytical Geometry? 🔗

Quote from ChatGPT: Analytical geometry, also known as coordinate geometry, is a branch of mathematics that combines principles of algebra and geometry. It involves studying geometric shapes and objects using a coordinate system.

Basically, the main concept of analytical geometry is to represent geometric figures and problems using algebraic equations and coordinate points. This distinction is relevant when asking things like « why the tangent of 90 degrees is undefined ».

Periodic functions? 🔗

A periodic function is a function that repeats itself at regular intervals.

In order for a function f:Domain->Codomain to be periodic, there has to exist a constant P>0 which satisfies the condition f(x+P)=f(x). Some notable examples of periodic functions are: sin, tg, the rotation of the Moon around the Earth, cos, etc.

The trigonometric functions 🔗

There are six basic functions used in trigonometry: sine, cosine, tangent, co-tangent, secant and cosecant. These functions represent ratios of sides in a right-angled triangle.

Sine, cosine, secant, and cosecant have a period of 2π while tangent and cotangent have a period of π.

For the angle ∢BAC of the triangle displayed below we define the trigonometric functions as follows:

$$ sin(\widehat{BAC}) = \frac{BC|Opposite} {AB|Hypotenuse}; cos(\widehat{BAC}) = \frac{AC|Adjacent} {AB|Hypotenuse} $$ $$ sec(\widehat{BAC}) = \frac{AB|Hypotenuse} {BC|Opposite}; csc(\widehat{BAC}) = \frac{AB|Hypotenuse} {AC|Adjacent} $$ $$ tg(\widehat{BAC}) = \frac{BC|Opposite} {AC|Adjacent}; ctg(\widehat{BAC}) = \frac{AC|Adjacent} {BC|Opposite} $$

One detail to keep in mind is that the trigonometric functions( in the case of Analytical Geometry) are always computed within a right triangle. Because of this, we can transpose stuff around and compute any of the trigonometric functions for angles greater than 180°. And because they are periodic, we can even go beyond 360°. Isn’t that exciting?!

If you want to understand this idea better you should check out this tool.

Speaking of angles… 🔗

Disclaimer: The following chapter is my interpretation of RADIANS AND DEGREES, the 4th chapter of Kalid Azad’s “Math, Better Explained”

Where do Degrees Come From? 🔗

«It’s an obvious fact that circles should have 360 degrees. Right?»

Well, kind of…

A circle having 360 degrees just feels natural. And it does so because it’s indeed kind of natural. 360 is an arbitrary number, An approximation of the Earth’s rotation around the Sun. This number was determined waay back then, when it was believed that one full rotation completes in drumroll 360 days. This value was backed up by astronomical observations and, to be honest, it’s impressive to think that it was determined using the technologies of 2000 BC.

Radians Rule, Degrees Drool 🔗

This subchapter is going to be just a copy paste (or put more beautifully, a quote), but me trying to adapt this would be just… Read the paragraph, okay?

« A degree is the amount I, an observer, need to tilt my head to see you, the mover. It’s a tad self-centered, don’t you think?

Suppose you saw your friend Bill running on a large track:

• Hey Bill, how far did you go?
• Well, I had a really good pace, I think I went 6 or 7 miles
• Shuddup. How far did I turn my head to see you move?
• What?
• I’ll use small words for you. Me in middle of track. You ran around. How… much… did… I turn… my… head?
• Jerk.

Selfish, right? That’s how we do math! We write equations in terms of “Hey, how far did I turn my head see that planet/pendulum/wheel move?”. I bet you’ve never bothered to think about the pendulum’s feelings, hopes and dreams. Do you think the equations of physics should be made simple for the mover or observer? »

Radians: The Unselfish Choice 🔗

Well, there is a way to make things simple! (Quote ahead:) «Instead of wondering how far we tilted our heads, consider how far the other person moved. Radians measure angles by the distance they travel. But absolute distance isn’t that useful, since, for example, going 10 miles is a different number of laps depending on the track. So we divide by radius to get a normalized angle:

$$ Radian = \frac{distance\ traveled} {radius} $$ A circle has 360 degrees or 2π radians — going all the way around is 2πr/r. So a radian is about 360/2π or 57.3 degrees.

“Great, another unit. 57.3 degrees is so weird.” Well, it is weird when you’re still thinking about you!

Radians are the empathetic way to do math — a shift from away from head tilting and towards the mover’s perspective. Strictly speaking, radians are a ratio (length divided by another length) and don’t have a dimension. Practically speaking, we’re not math robots, and it helps to think of radians as “distance traveled on a unit circle”.

The transition towards thinking in radians may be rough at first. But it shouldn’t! We encounter the concept of “mover’s distance” quite a bit:

• We use “rotations per minute” not “degrees per second” when measuring certain rotational speeds. This is a shift towards the mover’s reference point (”How many laps has it gone?”) and away from an arbitrary degree measure.
• When a satellite orbits the Earth, we understand its speed in “miles per hour”, not “degrees per hour”. Now divide by the distance to the satellite and you get the orbital speed in radians per hour.
• Sine, that wonderful function, is defined in terms of radians as

$$ sin(x) = x - \frac{x^3} {3!} + \frac{x^5} {5!} - \frac{x^7} {7!} \cdots $$ or more formally, $$ \sin x = \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} $$ This formula only works when x is in radians! Why? Well, sine is fundamentally related to distance moved, not head-tilting.»

So What’s the Point? 🔗

« Degrees have their place: in our own lives, we’re the focal point and want to see how things affect us. How much do I tilt my telescope, spin my snowboard, or turn my steering wheel?

With natural laws, we’re the observers describing the motion of others. Radians are about them, not us.

The two main points to take are:

• Degrees are arbitrary because they’re based on the sun (365 days ∼ 360 degrees), but they are backwards because they are from the observer’s perspective.
• Because radians are in terms of the mover, equations “click into place”. Converting rotational to linear speed is easy, and ideas like sin(x)/x make sense.»

Illustration 🔗

Here is an interactive illustrasion of the sin function. While writing the code I had a little “revelation” that instead of plotting cos on the X axis and sin on the Y axis I needed to do f(x,sin(x)), with x being the pixel position on the screen.

In order to draw the wave of the function, I needed to use 2 new factors: frequency (the value for which to compute a value) and amplitude (the constant for the X axis shift, needed for the graph’s amplitude). In the end the function became:

$$ f(x) = offset+amplitude*sin(x+frequency) $$