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Age 16 to 18
Challenge Level
Consider a right angled triangle with an acute angle of $\theta$. Let the base of the triangle be of length 2.
Find the height of the triangle in terms of $t$, where $t=\tan \theta$.
Now imagine a line in the triangle which forms an isosceles triangle with two angles equal to $\theta$.
Use this diagram to prove the double angle formula, where $t=\tan \theta$: $$\tan2\theta = {2t\over {1-t^2}}, \quad \sin2\theta = {2t\over {1+t^2}},\quad \cos2\theta = {{1-t^2}\over {1+t^2}}$$
NRICH is part of the family of activities in the Millennium Mathematics Project.