Further Maths - T-formulae

Pearson Edexcel Further Mathematics 2022

Created: October 09, 2021 | Read markdown | About these notes | View in context | Study these flashcards |

Flashcards

The standard t-formulae

What is the $t$-substitution ($t = ...$)?

t-substitution-definition
\[t = \tan\left(\frac{\theta}{2}\right)\]

What is $\sin\theta$ in terms of $t$?

t-formula-sin
\[\sin\theta = \frac{2t}{1 + t^2}\]

What is $\cos\theta$ in terms of $t$?

t-formula-cos
\[\cos\theta = \frac{1 - t^2}{1 + t^2}\]

What is $\tan\theta$ in terms of $t$?

t-formula-tan
\[\tan\theta = \frac{2t}{1 - t^2}\]

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

t-formulae-derivation-triangle

The alternative substitution t = tan(theta)

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

t-formula-sin-double-angle
\[t = \tan \theta\] \[\sin 2\theta = \frac{2t}{1 + t^2}\]

What trick should jump for proving

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

with a $t$-substitution?

t-substitution-pythagorean-identity-trick

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

Structure of the formulae and when to substitute

What is in the numerator for the $t$-formulae involving $\cos$ and $\sin$?

t-formulae-common-term
\[1 + t^2\]

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?

t-substitution-differentiate-before-substituting

Taking the derivative and then making the substitution.



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