Trigonometric identities

Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?

Can you find a way to prove the trig identities using a diagram?

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.

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

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.

Draw graphs of the sine and modulus functions and explain the humps.

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?

This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.

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.

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