Question

Explain the connection between angles and complex numbers we've been relying on. Illustrate
this by finding the sine and cosine of
1. A 22.5 degree angle, and
2. twice the angle in a 5-12-13 triangle opposite the side of length 5.

Answer 1

if we rename the axises of a conventional graph: Y to i, and X to Real; we have a representation of the complex plane. the complex number a+bi can then be positioned on the plane as (r,i) coordinants: r=a, i=b

any point on a plane can be defined by its distance and movement from the origin

Answer 2

ya i think i have this, can i ask you something though? for twice the angle, is it okay to use de moivre's theorem?

Answer 3

it sounds right but looks wrong i'm not quite sure...

Answer 4

not sure what DM looks like in this, but my interpretation of the question is something like this

Answer 5

given an abc right triangle
\[sin\frac{\alpha}{2}=\frac{c-b}{\sqrt{2c(c-b)}}\]
\[cos\frac{\alpha}{2}=\frac{a}{\sqrt{2c(c-b)}}\]
but i might have those mixed up