Sine, cosine, tangent

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

How far should the roof overhang to shade windows from the mid-day sun?

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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.

On a nine-point pegboard a band is stretched over 4 pegs in a "figure of 8" arrangement. How many different "figure of 8" arrangements can be made ?

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees before being reflected across to the opposite wall and so on until it hits the screen.

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?

What is the longest stick that can be carried horizontally along a narrow corridor and around a right-angled bend?