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

Trigonometry, circles and triangles combine in this short challenge.

How can you represent the curvature of a cylinder on a flat piece of paper?

What is an AC voltage? How much power does an AC power source supply?

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?

Two perpendicular lines lie across each other and the end points are joined to form a quadrilateral. Eight ratios are defined, three are given but five need to be found.

Two places are diametrically opposite each other on the same line of latitude. Compare the distances between them travelling along the line of latitude and travelling over the nearest pole.

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

Prove Pythagoras' Theorem for right-angled spherical triangles.