Sine, cosine, tangent

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

How would you design the tiering of seats in a stadium so that all spectators have a good view?

Can you find the sum of the squared sine values?

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

Trigonometry, circles and triangles combine in this short challenge.

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

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Without using a calculator, computer or tables find the exact values of cos36cos72 and also cos36 - cos72.

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?