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# $t$ for tan

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Age 16 to 18

Challenge Level

- Problem
- Student Solutions

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?

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

This comes in two parts, with the first being less fiendish than the second. Itâ€™s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.

You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.

This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.