Exact value for cos(2Pi/5) and sin(2Pi/5) using complex roots.

Complex roots can be used to calculate the exact trigonometric value for unusual angles; for example cos(2π/5) and sin(2π/5). Choosing z⁵=1 will produce five complex roots aligned as shown in the first diagram. All five roots must be equally spaced over 2π, so their radial spacing will be 2π/5.A right triangle can be superimposed on z₂ as shown in diagram 2 with the angle at the origin being 2π/5.

 

 

For z⁵=1 the modulus for all the roots =1, so z₁ … z₅ are all on a unit circle (diagram 3). Thus from trigonometry, the x-coordinate of z₂= cos(2π/5) and the y-coordinate of z₂ = sin(2π/5).

Calculating their value requires some clever maths.

 

In Cartesian form let z⁵=(x+yi)⁵.  If z⁵=1+0i, then (x+yi)⁵=1+0i. So the first step is to expand (x+yi)⁵ collect Re() and Im() parts and equate the Re() part to 1. (Diagram 4 lines 1 and 2)

 

This real part contains both x and y. In order to solve for x we first eliminate y by returning to the fact that z₂ is on a unit circle so x²+y²=1. Rearranging this gives y²=1-x² which can be used as a substitute for y² and y⁴ since y⁴=(y²)². (Diagram 4 lines 3, 4 and 5).

 

Now we have an equation in x only and resorting to a CAS calculator to solve it yields three exact values for x (Diagram 5 lines 1 and 2).

As z₂ is in the first quadrant we choose the positive value of x as cos(2π/5) and use this value to calculate y=sin(2π/5) (diagram 5 lines 3 and 4).