Mathematical Prelude - Trigonometry

Introduction

Often times, trigonometry is seen as a chore, a class that people take for the sake of it, yet many don’t realise that it’s importance. Trig is used in many fields from architecture, to engineering and even astrophysics! While this article doesn’t expect you to fully understand trig, I hope that you would at least understand the basics of what trig is and perhaps even solve a few problems. However before we even continue, we must first ask ourselves, what is trigonometry?

Can you imagine young people nowadays making a study of trigonometry for the fun of it? Well I did. — Clyde Tombaugh

Trigonometry is a branch of Mathematics that is used to study the relationship between angles and sides of triangles, while it does seem fairly elementary by definition, in reality, the subject tends to be more abstract and elegant. In fact, it helps to visualise Trigonometry not by using triangles directly but by using circles.

The Unit Circle

Say we have an x-y coordinate system and we draw a circle on this coordinate plane such that the centre of the circle is on the origin (0,0) and the circle has a radius of 1, we essentially have constructed something known as a unit circle.

A unit circle

If we draw a line from the origin to any point on the circumference of a circle, we have constructed a line with a length of 1. The equation for our circle is x²+y²=r² where r is the radius of the circle. As the radius of the unit circle is 1, we can simplify this to x²+y²=1. Now we’re going to start introducing trigonometry using the unit circle. Say we draw a line from the origin to a point on the circumference. This line is inclined at some angle ‘θ’

The lines end at a point P with coordinates (a,b). Essentially what has happened is that we have travelled along the x axis by a distance a and travelled along the y axis by a distance b.

Now we’re going to digress a bit here, so bear with me but we need to first of all understand what the trigonometric functions are in the first place, essentially what it does is it takes in angle and spits out an output. The three trigonometric functions are sine, cosine and tangent and they can be remembered using this acronym: SohCahToa. This essentially states the sides of the angle used for a specific trigonometric function. S, C and T stand for sine (sin), cosine (cos) and tangent (tan) respectively. o, a and h stand for opposite, adjacent and hypotenuse respectively. Essentially for our triangle in the unit circle, the angle is θ, the opposite side is b (this side is opposite to the angle), the adjacent side is a (this side is adjacent to the angle) and the hypotenuse is the longest side of the triangle, for the unit circle, the hypotenuse is equal to the radius of the circle and thus has a length of 1. The trigonometric functions can be expressed like this:

Now if we go ahead and evaluate this for our unit circle we get:

You can also express tan using the following formula:

Note that this only works for the unit circle, you would have different values if you used an actual triangle

So how does trigonometry actually work? Well let’s suppose you knew that θ was 45 degrees, what would b equal? Well you know your value of h and you want to find out the value of o, you would use sine. Therefore

Now you would be able to find the length of a either by using cosine or Pythagoras Theorem.

Radians

We’ve been working in degrees this entire time but Physicists prefer using another unit to measure angles known as Radians. It is effectively based on the unit circle, the circumference of the unit circle would be 2πr or 2π. Now if we chose an arc in the circle, such that it was equal to the radius, in this case an arc length of 1. It would form an angle like so

Notice how if we wanted to cover the entire circumference with 1 unit arcs, we would need 2π arcs? Therefore if we wanted to cover the entire circle, we would essentially be covering 360 degrees as well. Therefore 2π radians = 360 degrees. We can use this to get the following angles:

2π radians = 360 degrees.

π radians = 180 degrees

π/2 radians = 90 degrees

As such, the formula for converting between degrees and radians is given by the following:

Special Angles

If you were to input angles or radians into the trigonometric function, it looks like you get a random answer, usually long decimals, however some special angles are worth remembering as they usually give you a clean answer

Special angles

Graphs

Each trig function can also be graphed where the x axis takes on the different angles in radians and the y axis takes on the output

Graph of sin(x)

Graph of cos(x)

Graph of tan(x)

Graphs or waves that look like the graph of sin(x) are known as sinusoidal

Trigonometry and Astrophysics

We’ve just looked at a lot of trigonometry… but how does it come into play for Astrophysics? Trigonometry is mainly used to calculate distances to stars via a method known as the “Trigonometric Parallax method”, we’ll get into what this method actually is very soon, we just have some Maths to look at.

I hope you guys enjoyed this brief introduction into trigonometry, trigonometry has it’s uses in many different fields and astrophysics is no different. In the next article, we’ll be looking at vectors and scalars!

Sources used:

1. Jupyter Notebook and LaTeX
2. Desmos Graphing Calculator