Golden rato 3: Φ and sin(54)

Geometry Level 2

What is the relationship between $Φ$ and $\sin(54°)$ ? Note: $Φ$ represents the golden ratio. Do not use a calculator.

Φ = 1/sin(54) Φ = - sin(54) Φ = 2 sin(54) Φ = sin(54) / 3

1 solution

Yashas Ravi
Apr 1, 2018

$\sin(54) = \cos(90-54) = \cos(36)$ . Since $Φ =$ $\frac{√5+1}{2}$ and $\cos(36) =$ $\frac{√5+1}{4}$ , multiplying $\frac{√5+1}{4}$ by 2 gives us $\frac{√5+1}{2}$ , so $Φ = 2\cos(36)$ or $Φ = 2\sin(54)$ . For the derivation of why $\cos(36) =$ $\frac{√5+1}{4}$ , see this link: Cos(36) derivation

Golden rato 3: Φ and sin(54)

1 solution

0 pending reports