# Further Maths - T-formulae

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/t-substitution/ · Updated: 2021-10-09 · Tags: school, further-maths, further-pure-1

## 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](t-formulae-deriv.png)

### 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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