"Explain what the connection between angles and complex numbers says about the relation between (cos(x), sin(x)) and (cos(x + pi/2), sin (x + pi/2)). Is rotation or angle addition somehow involved? By what angle? What complex number corresponds to this angle?"

I know that sin=y and cos=x, and that sin(x)/cos(x) = tan(x), which equals the slope (y/x). I know that the point where the line intersects the Unit Circle is called the terminal point (x,y) aka (cos(x), sin(x)), and that the complex number for that is (a + bi).

Can someone explain to me how to go about doing this?

Question

"Explain what the connection between angles and complex numbers says about the relation between (cos(x), sin(x)) and (cos(x + pi/2), sin (x + pi/2)). Is rotation or angle addition somehow involved? By what angle? What complex number corresponds to this angle?"

I know that sin=y and cos=x, and that sin(x)/cos(x) = tan(x), which equals the slope (y/x). I know that the point where the line intersects the Unit Circle is called the terminal point (x,y) aka (cos(x), sin(x)), and that the complex number for that is (a + bi). 

Can someone explain to me how to go about doing this?

kainui · Answer

Well if you draw out a complex number in the plane, it looks like this: |dw:1402854518470:dw| So you can see that a=cosx and b=sinx.

So equivalently you can write out cosx+isinx as the complex number. Then you can find that angle using arctan(b/a) and find the distance from the origin to the end as the hypotenuse of a right triangle by the pythagorean theorem. That way you've gone from describing it with a & b to describing it with an angle and radius.

anonymous · Answer

Ok, I understand everything except for arctan(b/a). I am familiar with tan(x), but my professor hasn't gone over arctan before. Can you explain that to me?