Sine, cosine, tangent

problem

Orbiting Billiard Balls

Age

14 to 16

Challenge level

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

Pythagoras on a Sphere

Age

16 to 18

Challenge level

Prove Pythagoras' Theorem for right-angled spherical triangles.
problem

Dodecawhat

Age

14 to 16

Challenge level

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.
problem

Moving Squares

Age

14 to 16

Challenge level

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

Six Discs

Age

14 to 16

Challenge level

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?
problem

Stadium Sightline

Age

14 to 18

Challenge level

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

Degree Ceremony

Age

16 to 18

Challenge level

Can you find the sum of the squared sine values?
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

Ball Bearings

Age

16 to 18

Challenge level

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.
problem

Gold Again

Age

16 to 18

Challenge level

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