Trigonometry and Pythagoras - algebra

Trigonometry and Pythagoras

Trigonometry and Pythagoras

I want to focus in this post on the differences between trigonometry and pythagoras. When to use each one and so on. Let’s start by reminding ourselves of the equations you will hopefully be familiar with by now, presuming you have learnt a little bit about both.


Sin, Cos and TanTrigonometry and Pythagoras - algebra

This will always be involving three buttons on your scientific calculator:

Sin, cos and tan. These are not values, but functions. They do something to the numbers you put into them. They are like machines.

sin 30 is often written as sin(30) to emphasise this point. It does not mean ‘sin multiplied by 30’. That doesn’t make any sense. It would be like saying ‘plus multiplied by 30’. We’ll not worry too much about why sin(30) =  0.5 or sin(90) = 1 here. We’ll just trust that something is happening inside of our calculator that we don’t understand, but helps us to solve the problems.



Soh Cah Toa Vs Pythagoras


Let’s take a look at three problems…

Trigonometry and Pythagoras - calculator


In problem 1, we have an angle and a side, and we need to find the length of another side. In problem 2, we have 2 sides and we need to find the size of one of the angles.

But, in both problems we are working with 2 sides and one angle. This is the key. Don’t think “What do I have?” Think “What am I working with?” If we’re working with angles and sides it’s trigonometry.

In problem 3 we don’t have any angles and we don’t want to know any angles. So we are only working with 3 sides. If we are only working with sides, this means its pythagoras.


How to label the triangle when using Soh Cah Toa

Trigonometry and Pythagoras - Problem 1

When I mention trigonometry, I really mean what is often called soh cah toa. There is a more advanced type of trigonometry involving sin, cos and tan, but we will probably look at that in a different post. Let’s take another look at problem 1.





We need to label the triangle first with the following labels:

o for the side that is opposite the angle we are working with.

a for the side that is adjacent (next) to the angle we are working with.

h for the hypotenuse. This is the side that is opposite to the right angle and is always the longest side on a right angled triangle. 


We can also label the angle we are working with using a greek letter, θ, pronounced theta.


See the diagram below:

Trigonometry and Pythagoras - hypotenuse

Applied to problem 1, it would look like this:

Trigonometry and Pythagoras - adjacent

Problem 1 is asking us to calculate the adjacent side.


Using Soh Cah Toa


The trick to using this is to raise the middle letter each time you write out soh cah toa as below.

Trigonometry and Pythagoras - soh cah toa

We can then draw a triangle around each one and we have instant formula triangles to work with. I’ve also colour coded the letters the same colours as in the above diagram.

Trigonometry and Pythagoras - soh cah toa2

s = sin(θ)

c = cos(θ)

t = tan(θ)

In our question we are working with o and a, that sounds like toa, so we will be using the 3rd formula triangle. 

Trigonometry and Pythagoras - toa


We want the adjacent side, so covering that up we can see that:

Trigonometry and Pythagoras - diagram


So RT = 14cm ÷ tan (53°) = 10.5cm (1dp)


What if we need to find out an angle?

Trigonometry and Pythagoras - Problem 2


In problem 2 it’s an angle we need to find, not a side. Using the same technique as described above we can see that…

Trigonometry and Pythagoras - soh

Sin (ACB) = 8cm ÷ 10.5cm = 0.762 (3sf)


But… This is not the angle ACB. This is just the sin of the angle. We need to put it through the sin machine backwards to ‘un-sin’ it! To do that we use an inverse-sin (sin-1). The same would be the case with tan and cos. You can access the inverse sin tapping ‘shift’ and then the pressing sin



Angle ACB = Sin -1(8cm ÷ 10.5cm) = 49.63° (2dp)


And finally, problem 3!

Let’s briefly go over


In problem 3 we are not interested in any angles and so we’re going to use pythagoras. Let’s briefly look at how to label a triangle with pythagoras… Because it’s different!

Trigonometry and Pythagoras - Problem 3a

This time the hypotenuse is labelled as c and the other two sides are a and b. It doesn’t matter which way around you label a and b, but c is always the longest side, opposite the right angle. You can also see the famous Pythagoras theorem as well.

Notice we have another a, but this does not mean adjacent. What would it be adjacent to? We are not working with angles this time! It’s just a.


Ok, in our case it is the hypotenuse we are trying to find out. So…

AB2 = 152 + 82 = 225 + 64 = 289.

Trigonometry and Pythagoras - Problem 3

But… This is AB2, not AB. So I need to square root it.

Trigonometry and Pythagoras - Problem 3b



So there you have it. 3 problems. Problem 1 using soh cah toa with a missing side. In Problem 2, we had to use an inverse trigonometry function. Problem 3 was pythagoras. I hope you found that helpful and I’d suggest searching the internet for a few practice questions with answers in these topics to see if you are able to do them now.







A bit about the author, Paul H:

Paul is a qualified and experienced Physics, Maths, and Science teacher, now working as a full-time tutor, providing online tuition using a variety of hi-tech resources to provide engaging and interesting lessons.  He covers Physics, Chemistry, Biology, and Science from Prep and Key Stage 3 through to GCSE and IGCSE. He also teaches Physics, Maths, and Chemistry to A-Level across all the major Exam Boards.

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