Let's split them up into two separate complex numbers to start with, one being the numerator, 2i+52i+5, and one the denominator, -7i+7−7i+7.
We want to get them from linear (x+iyx+iy) form to trigonometric (r(costheta + isintheta)r(cosθ+isinθ) where thetaθ is the argument and rr is the modulus.
For 2i+52i+5 we get
r = sqrt(2^2 + 5^2) = sqrt29r=√22+52=√29
tantheta = 2/5 -> theta = arctan(2/5) = 0.38 " rad"tanθ=25→θ=arctan(25)=0.38 rad
and for -7i+7−7i+7 we get
r = sqrt((-7)^2+7^2) = 7sqrt2r=√(−7)2+72=7√2
Working out the argument for the second one is more difficult, because it has to be between -pi−π and piπ. We know that -7i+7−7i+7 must be in the fourth quadrant, so it will have a negative value from -pi/2 < theta < 0−π2<θ<0.
That means we can figure it out simply by
-tan(theta) = 7/7 = 1 -> theta = arctan(-1) = -0.79 " rad"−tan(θ)=77=1→θ=arctan(−1)=−0.79 rad
So now we've got the complex number overall of
(2i+5)/(-7i+7) = (sqrt29(cos(0.38)+isin(0.38)))/(7sqrt2(cos(-0.79)+isin(-0.79)))2i+5−7i+7=√29(cos(0.38)+isin(0.38))7√2(cos(−0.79)+isin(−0.79))
We know that when we have trigonometric forms, we divide the moduli and subtract the arguments, so we end up with
z = (sqrt29/(7sqrt2))(cos(0.38+0.79)+isin(0.38+0.79))z=(√297√2)(cos(0.38+0.79)+isin(0.38+0.79))
= 0.54(cos(1.17)+isin(1.17))=0.54(cos(1.17)+isin(1.17))
