Building blocks

# $t$ for $\tan$ Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Suggestion

If we write $t = \tan \theta$, then the following equations are true.

\begin{align*} \tan 2\theta &= \frac{2t}{1-t^2}, \\ \sin 2\theta &= \frac{2t}{1+t^2}, \\ \cos 2\theta &= \frac{1-t^2}{1+t^2}. \end{align*}

Can you use this diagram to obtain these formulae?

What lengths and angles might be useful to us here? Can you find the missing lengths and angles in terms of $t$?

Which angle is $2\theta$?

For what range of values of $\theta$ does this argument work?

The relationships are shown using a triangle. Are there any values of $\theta$ for which the diagram does not work? Can you adapt it?