In this post I’ll be proving to you that:
Now firstly I will have to say that:
And also that:
If this is the case, then…
Since this is in the form:
I would have to conclude that:
Hence I’ve proven that:
In this post I’ll be showing you how to prove that:
Firstly, let’s say that:
If this is the case, then according to the rules of complex numbers:
Secondly, let’s determine what is…
As you can see, we get the result above – which is another complex number.
This means that:
Therefore we’ve proven that:
You can watch a video related to this proof below…
In this post I’m going to be proving that…
So, here I go…
Wait for it…
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! 😀
So, you have the equation of the ellipse but you need to completely isolate y. How would you go about doing this? Well, here is a fantastic example…
This will come in handy if you’re trying to derive the area of an ellipse from absolute scratch.
Thanks for reading! 🙂
In this post, I’ll be demonstrating how one can derive the formula for an ellipse from absolute scratch.
To derive the formula for an ellipse, what we must first do is create a diagram like the one below.
** Click on the image above to see it in full size.
Now, the first thing we’ve got to acknowledge here is that:
What we’re basically saying is that D_1 + D_2 is equal to the length from -a to a in the diagram above.
This formula can be understood by watching the video below…
These photographs can also help the formula sink into your mind…
Ellipse Image 1:
Ellipse Image 2:
Now, look at the diagram at the top of this page once again…
What you will notice is that:
If this is the case, we can say that:
** Click on the image of the workings to see it in full size.
Alright, so far so good… Now, it turns out – if you look at the diagram at the top of this page carefully, you will discover that:
And this ultimately means that:
The formula you see just above is the formula for an ellipse. You’ve derived it from scratch!!
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:
And if we use the trigonometric identity below:
We will reach the conclusion:
Now, sin(A+B) as a trigonometric identity, is:
And, thanks to SOH CAH TOA…
Which means that…
As this is the case, we can conclude that:
Recently I discovered a few more proofs, some related to A Level mathematics. You can access these proofs by clicking on the links below.
Derive the formula of an ellipse: https://plus.google.com/+mathsvideosforweb/posts/ZF8D3ghSNRD
Discover the distance between point A and B on the edge of a circle: https://plus.google.com/+mathsvideosforweb/posts/9fmuEJ4qeXY
Find the area of a sector within a circle: https://plus.google.com/+mathsvideosforweb/posts/cHt9t9PeSfg
Formula for a torus, derived from scratch: https://plus.google.com/+mathsvideosforweb/posts/TS84TL44BYd
Create the mathematical singularity shown in movie documentaries: https://plus.google.com/+mathsvideosforweb/posts/7p2onSzNYte
Parameterised formula for a torus derived from scratch: https://plus.google.com/+mathsvideosforweb/posts/AZrVMiPVdbv
Derive the formula for an ellipsoid (normal and parameterised) from scratch: https://plus.google.com/+mathsvideosforweb/posts/Wijf94ztn3v
If you love mathematical proofs, please feel free to join my ‘mathematics proofs’ Google community at: https://plus.google.com/communities/106007058741903558109
I’ve also got a new Facebook page related to stuff about the universe and also mathematics. You can add me as a friend by accessing this link: http://www.facebook.com/tiago.hands
DOWNLOAD FREE A LEVEL MATHEMATICS PROOFS:
Today you can download and view my hand written A Level maths proofs. These hand written proofs can be found at: https://plus.google.com/communities/106007058741903558109
If you do decide to share my work, please make sure my web links remain intact so that I can continue to build and re-invest in mathsvideos.net.
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.
Thankyou for your attention. 🙂