The golden ratio comes up in unexpected places. This post will show how the sines and cosines of some special angles can be expressed in terms of the golden ratio and its complement.
Recall the golden ratio is
φ = (1 + √ 5)/2
and the complementary golden ratio is
φ’ = (1 – √ 5)/2.
The derivation begins by solving the trigonometric equation
sin 2θ = cos 3θ
in two different ways. To make the solution unique, we look for the smallest positive solution.
First, note that the sine of an angle is the cosine of its complement, i.e.
sin(x) = cos(π/2 – x).
So our equation can be written as
cos(π/2 – 2θ) = cos 3θ.
The smallest positive solution satisfies π/2 – 2θ = 3θ, and so θ = π/10 or 18°.
Now let’s solve the same equation another way. First, we use the double and triple angle identities.
sin 2θ = 2 sin θ cos θ
cos 3θ = 4 cos3 θ – 3 cos θ
Set the two equations above equal to each other and divide by cos θ. Then we have
4 cos2 θ – 3 = 2 sin θ.
Substitute 1 – sin2 θ for cos2 θ and the result is a quadratic equation in sin θ:
4 sin2 θ + 2 sin θ – 1 = 0.
From the quadratic equation, the solutions are sin θ = (-1 ± √ 5)/4. The positive solution is
sin θ = (-1 + √ 5)/2 = -φ’/2.
Setting the solutions obtained from both methods equal to each other,
sin π/10 = sin 18°= -φ’/2.
We can now use common trig identities and the above result to express the sines and cosines of other angles in terms of φ. Switching to degrees will make the following a little easier to read.
We know sin 18° = -φ’/2, and so cos 72° = -φ’/2. We can use the sum angle identities to express the sine and cosine of every multiple of 18° in terms of φ. Also, we could apply the half angle identities to express the sine of cosine of 9° in terms of φ, and then again by addition formula we could extend this to all multiples of 9°.
This post was an expanded form of a derivation given in The Divine Proportion.