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.

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Best wishes for your study and your mathematics!

[MrVoayer]

Great example of solving inocus looking trigonometric equation using t-Formulae ! Yet another reason to learn t-Formulae by heart !

[tinker]

Amazing videos! Really inspiring, stay strong and keep up the brill videos <3

[thomasevans]

thank you very much mate, all the way from london in 2019

[ijeremyoliver]

Prof: (uses function inverse notation instead of arctan, reports solutions in minutes and seconds)
Me: Australia is a lawless place.

[drbsr]

Hi,
I got the correct answer on my little casio fx-260 solar calculator, however when I entered tan-1(-2+sq root 7 /3 into my Ti-84 graphing calculator I got -48.1909... for an answer
? probably a setup error on my part. however I tried different mode settings with no luck . Can you guide me or instruct me on how to correctly enter this formula into the TI-84 Plus Silver Edition Calculator. Thanks

[oppongedmond]

I had 12.14759447 for the tan-1 on my Casio fx 991 es calculator