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Gold Again

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

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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.

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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 Construction

Age 16 to 18 Challenge Level:

Just do part 1 if you like where you will construct a golden rectangle and find an approximate value of the golden ratio by measurement.

If you want to go further there are many hints within this question to take you through it. You will find the value of $\phi$ first by drawing and measurement, then by using geometry and calculating with surds (square roots), then by taking a reading from a graph and finally by rearranging a formula into a quadratic equation and solving it.