Gold Again

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

Pythagorean Golden Means

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

Golden Triangle

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

Golden Ratio

Stage: 5 Challenge Level:

Take $b = 1$ and write the equation in terms of $a$. Then solve this equation to find the golden ratio.