Complex numbers in mod-arg form ( cis )

[TheAkuma]

Complex numbers in mod-arg form ("cis")

Greetings, I'm learning about the mod-arg form. I find it fairly easy when I come across simple radians that relate to the two special triangles like pie/3, pie/4 and pie/6. But when the radians become a little bit more complicated like 3pie/4 I'm in the foetal position. This equation in particular; "2cis3pie/4"

I only get to 2cos 135degress + i2 sin 135degrees

I can't simplify it down. I know that 3pie/4 is 135degrees but i can't convert it to a fraction. If anyone could help me with a method to solve more complicated radians that would be much appreciated.

 

[LeonhardEuler]

One way is to use the sum and difference formulas for sine and cosine:
cos(135) = cos(180 - 45) = ?

 

[Dick]

You can either look at a graph of sin(x) and cos(x) and realize that 3*pi/4 has a lot in common with pi/4 using symmetries, or you can realize 135=90+45 and use addition formulas like sin(a+b)=sin(a)*cos(b)+cos(a)*sin(b) etc. I.e. sin(a+90)=cos(a), so sin(135)=cos(45).

 

[TheAkuma]

Ohh! ok. So I can use the sum in this case. It's confusing for me cause I didn't now if I'm allowed to use the sum of cosine or sine of the angle since I do two math subjects and I'm not allowed to use some methods in one maths subject. So I can also draw an Argand Diagram to help me out as well? I think I should just stick to the graph (argand diagram) since that leans towards the maths subject I am doing. so would I draw it as 135degrees on one side of the graph then 45degrees in in the special triangle on the otherside? Or am I completely off track?

 

[Dick]

Yes, just use the graph. The special triangles are symmetrical.

 

[TheAkuma]

ok thanks