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# Double Angle and Half Angle Formulae (and Use of t-Formulae)

See below

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.

t-Formulae ~ 02 ~ Used to Solve Trigonometric Equations

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

Because t-formulae allow us to convert trigonometric functions into fractions involving polynomials, we can use them to change trigonometric equations to polynomial equations. This can often (but not always) be an advantage, because mathematicians have developed some amazing tools when it comes to solving polynomial equations.

Such substitutions are particularly valuable when the original equation only involves single instances of trigonometric functions and no powers. In fact, this is when t-formulae are almost exclusively used. In other words, we look for equations that involve uses of sin(x), cos(x), and tan(x), but not powers of these functions.

In this video, I provide a 'typical' example ... cos(x) = sin(x) + 1/2 ... and show how the use of t-formulae allows us to create a quadratic equation and then deduce all the solutions.

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

t-Formulae ~ 03 ~ Used to Evaluate Certain Integrals

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

Because t-formulae allow us to convert trigonometric functions into fractions involving polynomials, we can use them to change certain trigonometric integrals to 'integrals of fractions containing polynomials.' This can often (but not always) be an advantage.

Such substitutions are particularly valuable when the original integral only involves single instances of trigonometric functions and no powers. In fact, this is when t-formulae are almost exclusively used. In other words, we look for integrals that involve uses of sin(x), cos(x), and tan(x), but not powers of these functions.

The fraction that results will contain two polynomials. These may produce a logarithmic structure, or may require a division to take place, or may lead to the use of partial fractions, or may even produce inverse tangent functions, for example ... but many of these structures are almost instantly recognisable and experienced mathematicians know how to 'handle' them.

In this video, I provide a 'typical' example ... ∫1/(3 + 5cosx).dx ... which is Jim Coroneos' 30th integral (see my YouTube playlist, https://www.youtube.com/playlist?list=PLJ0RnXkLrsTRPyurk9ZsAQM0atYAKjsPB).

Because the complete solution of this integral has been accomplished in another video, I only spend sufficient time with the integral to show how the use of t-formulae allows us to convert it to a much more useable and recognisable form.

I had the privilege of being one of Graeme’s students in 2005. As a creatively minded individual I struggled greatly with Mathematics and was falling behind in my HSC studies. Under his careful and creative tutelage I began to truly understand Mathematics for the first time in my life. Graeme patiently taught me to master mathematical concepts in ways I could understand. In the week intensive I spent with him he taught me the entire HSC course, and I went on to receive a Band 6. I cannot recommend him highly enough.

Roxanne G (student, 2005)

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