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.
A short introduction to complex numbers written primarily for students aged 14 to 19.
Find out how the quaternion function G(v) = qvq^-1 gives a simple
algebraic method for working with rotations in 3-space.
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.
Can you find a way to prove the trig identities using a diagram?
Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?
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.
Can you explain what is happening and account for the values being
This polar equation is a quadratic. Plot the graph given by each
factor to draw the flower.
Draw graphs of the sine and modulus functions and explain the
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.
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0
what can you say about the triangle?
Join some regular octahedra, face touching face and one vertex of
each meeting at a point. How many octahedra can you fit around this
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. . . .