I have been given a question, one that should be fairly simple but I cant seem to figure this one out and not to sure where to start. The question is to solve the equation z^4 = -1 where z is a complex variable. They also state that modulus argument form is needed to solve this. I know this is |z| which is equal to 1 obviously but i don't know where to go from there. I know the answer is +- (1/sqrt(2)) +- (1/sqrt(2))i but that's cause they gave us the answer. i understand that if you apply the modulus form to either of the answers you get 1. What i don't understand is how to get to the answer from z^4 = -1. Not to sure if we're supposed to just see that or if there is a more practical way of solving this that will help me to make sense of it
"They also state that modulus argument form is needed to solve this". No idea what this means, so this may not be apropos. Sometimes, teachers will call it a "wrong" answer, even if it's right, but you solved it in a manner they didn't like. SUX, but what can you do?
The key to solving: z^4= -1 is to remember that: -1= 1.0/_180deg (Steinmetz representation where it's understood that this stands for: 1.0 X e^(j * pi) and Euler's Constant and j are understood, and that degrees are preferred since they're whole numbers, but that the only arguments to trig functions are in radians.)
Since it's a fourth order polynomial, we need four answers. To get them, modify accordingly:
z^4= 1.0/_(180 + 360n) Going 360deg (2pi rad) always brings you right back where you started. Then extract the fourth root of the right side of the equation:
z= [1.0/_(180 + 360n)]^0.25
1.0^0.25= 1.0
180*0.25= 45
360*0.25= 90
z= 1.0/_(45 + 90n)
z_0= 1.0/_45= (1/sqrt(2)) + j(1/sqrt(2))
z_1= 1.0/_(45 + 90)= -(1/sqer(2)) + j(1/sqrt(2))
z_2= 1.0/_(45 + 180)= -(1/sqrt(2)) - j(1/sqrt(2))
z_3= 1.0/_(45 + 270)= (1/sqrt(2)) - j(1/sqrt(2))
Four answers, and done. If you need this in Cartesian form, you can extract the answer via Euler's Identity, or get the answer from a scientific calculator. You just need to figure once, since these answers appear as complex conjugates.