Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

Why Stop at Three by One

Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.

Reflect Again

Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.

T for Tan

Age 16 to 18Challenge 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 $t=\tan \theta$: $$\tan2\theta = {2t\over {1-t^2}}, \quad \sin2\theta = {2t\over {1+t^2}},\quad \cos2\theta = {{1-t^2}\over {1+t^2}}$$