t-Formulae ~ 01 ~ Three Ways to Derive Them

One of my subscribers asked me to explain what t-formulae are, and how they are used in trigonometry. This is the first of three videos that I created in response to that request (thank you, Kristyn).

I firmly believe that it is a good idea to practise deriving significant formulae and identities as part of your regular study. In this way, you become intimately acquainted with each 'formula' and can also see how it is connected with the foundational concepts for that topic/area of study. You also learn basic methods for manipulating the symbols algebraically.

Therefore, this first video is devoted to demonstrating three different ways in which you could derive the t-Formulae. If you wish to skip to each method of derivation here are the key sections of the video:
02:05 ~ Explanation of what t-Formulae ARE
03:55 ~ Derivation #01 ~ Using Double-Angle Formulae
12:40 ~ Derivation #02 ~ Using a 'Half-Angle' Triangle
18:10 ~ Derivation #03 ~ Using a 'Full'-Angled Triangle

I suggest that you adopt the method that you like most and practise using it until you can derive the t-Formulae with ease!

In the second video I will demonstrate how t-formulae can be used to solve certain kinds of trigonometric equations.

In the third video I will explain how the t-formulae can help you evaluate particular kinds of integrals.

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Best wishes for your study and your mathematics!

[rosette_renah]

This is a beautiful video and the only video I could see and use for the proof. Thanks for making the proof Crystal Clear😉😉

[uachannel]

So much helpful, you've illuminated my mind on this topic. Thank you!

[UzairAhsanWarsi]

This has been very helpful Sir. Thanks.

[theskoomacat]

Hello again Graeme, thanks for posting these videos!
Nowadays I've been busy with tons of things so I hadn't had much time for maths, but after this week, I'll be jumping head into it. I'm currently working on a task, given by my PhysChem teacher, and that is deriving Heron's formula of triangle area from thin air. I've been battling with it for a while, trying all sorts of trigonometric laws and all that. I hope I'll be solving the problem soon.
Best wishes, Steve.

[MrVoayer]

As these t-Formulae refer to trigonometric functions somehow I find the derivation #2 usig half angle triangle the most accessible and the easiest one to understand. Nevertheless, great job, Graeme ! Valuable explanation of some very, very useful formulae !

[ДаняКурсинский]

So great, appreciate very much what you are doing!!!

[muskamelogbi]

When you decide to travel, firt of all you beguin, step by step, then the way appear, it 's your own way, maybe a sky's gift accompan you for a while, but no one will acheive your goal instead of you. thanks lot my friend.

[holyshit]

6:57 In my opinion it would be better if you divide it by one but this one should come from Pythagorean identity