This expression for the golden ratio is quite common, however, before I produced this post – I think it would’ve been very hard to figure out how to derive it from scratch. There aren’t many quirky proofs like this one on the internet – I am quite certain. I hope you liked reading this post! 😀
In this post I’ll be demonstrating how one can derive the three formulas which can be used to find the areas of triangles.
These formulas are in fact:
To begin with, let’s start by looking at the diagram below:
Now, if you look at the diagram carefully – you will notice that the area of the triangle is:
This can be simplified into:
Because of SOH CAH TOA, what we can also say is that:
This ultimately means that:
Alright, so far so good… Now we must put the icing on the cake and attach the final piece of the jigsaw puzzle to the formula above. In order to find the three equations which can be used to find the areas of triangles, we must now discover the expression for sin(C). We can discover its expression by first saying that:
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As always, I strive to produce the clearest A Level maths proofs on the internet, so that you can pass your C1, C2, C3 and C4 exams with flying colours. I’m not the kind of tutor who simply tells students to plug numbers into formulas. I make sure my students understand the problems they’re dealing with – using proofs and logical models.