Further Maths - T-formulae

Created: October 09, 2021

Flashcards

2021-10-09

What is the $t$-substitution ($t = …$)?
\[t = \tan^{-1}\left(\frac{\theta}{2}\right)\]

What is $\sin\theta$ in terms of $t$?
\[\sin\theta = \frac{2t}{1 + t^2}\]

What is $\cos\theta$ in terms of $t$?
\[\cos\theta = \frac{1 + t^2}{1 - t^2}\]

What is $\tan\theta$ in terms of $t$?
\[\tan\theta = \frac{2t}{1 - t^2}\]

What triangle can you imagine for deriving the $t$-formulae?

PHOTO T FORMULAE DERIV

2021-10-12

What $t$-substitution could you make other than $t = \tan\left(\frac{\theta}{2}\right)$ in order to rewrite $\sin 2\theta$?
\[t = \tan \theta\] \[\sin 2\theta = \frac{1 + t^2}{2t}\]

What trick should jump for proving

\[\sin^2 2\theta + \cos^2 2\theta = 1\]

with a $t$-substitution?

Using $t = \tan\theta$ rather than $t = \tan\left(\frac{\theta}{2}\right)$.

2022-01-20

What is in the numerator for the $t$-formulae involving $\cos$ and $\sin$?
\[1 + t^2\]

If

\[P(x) = 105 - 20\sin(6x) + 4\cos(12x)\]

and you were asked to make a substitution and find the derivative, what would be easier: taking the derivative and then making the substitution, or doing the substitution and then taking the derivative?

Taking the derivative and then making the substitution.



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