Trigonometric identities

article
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.
article
What are complex numbers?

This article introduces complex numbers, brings together into one bigger 'picture' some closely related elementary ideas like vectors and the exponential and trigonometric functions and their derivatives and proves that e^(i pi)= -1.
interactivity
Round and round a circle

Age

14 to 16

Challenge level

Can you explain what is happening and account for the values being displayed?
problem
Octa-flower

Age

16 to 18

Challenge level

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?
problem
Shape and territory

Age

16 to 18

Challenge level

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
problem
Sine and cosine for connected angles

Age

14 to 16

Challenge level

The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it.
problem
Reflect again

Age

16 to 18

Challenge level

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.
problem
Quaternions and rotations

Age

16 to 18

Challenge level

Find out how the quaternion function G(v) = qvq^-1 gives a simple algebraic method for working with rotations in 3-space.
problem
Quaternions and reflections

Age

16 to 18

Challenge level

See how 4 dimensional quaternions involve vectors in 3-space and how the quaternion function F(v) = nvn gives a simple algebraic method of working with reflections in planes in 3-space.
problem
Three by one

Age

16 to 18

Challenge level

There are many different methods to solve this geometrical problem - how many can you find?