Sine, cosine, tangent

In this 'mesh' of sine graphs, one of the graphs is the graph of the sine function. Find the equations of the other graphs to reproduce the pattern.

From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.

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

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?

The Earth is further from the Sun than Venus, but how much further? Twice as far? Ten times?

The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?

Trigonometry, circles and triangles combine in this short challenge.

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

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