Trigonometric identities

problem
Trig reps

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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

Age

16 to 18

Challenge level

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