We are trying to get it to the form; z=r(costheta+isintheta)z=r(cosθ+isinθ), usually called z=rcisthetaz=rcisθ
Where;
The complex number is z=a+biz=a+bi
The modulus (or absolute value) of the complex number zz is r=|z|=sqrt(a^2+b^2)r=|z|=√a2+b2
And tantheta=b/atanθ=ba
Now to the question, first we find the modulus of |z||z| or rr
z=2-2iz=2−2i
|z|=r=sqrt(2^2+(-2^2))=sqrt(4+4)=sqrt8=2sqrt2|z|=r=√22+(−22)=√4+4=√8=2√2
Let's find thetaθ
tantheta=(-2)/2=-1tanθ=−22=−1
We know that tantheta=(opposite)/(adjacent)=sintheta/costhetatanθ=oppositeadjacent=sinθcosθ,
when r=1r=1 for the latter.
If you sketch a right angled triangle and put -2−2 opposite thetaθ and 22 adjacent to thetaθ, the hypotenuse will be -sqrt2−√2.
If you are familiar with the unit circle, you will know that the reference angle for the triangle is 45° or pi/4.
If you are not familiar with the unit circle, using the triangle drawn,
we know this because cos=(adjacent)/(hypoten use)=sqrt2/2 and
sin=(opposite)/(hypoten use)=sqrt2/2 which are the values for 45°
Here though the tangent of 45° or pi/4 is 1 and not -1.
How do we figure out what gives us a tangent of -1?
Easy! We know that a=2 and b=-2.
If a, the abscissa (what we normally refer to as the x- coordinate) is positive, then it means we are looking at Quadrant I or IV.
If b, the ordinate (what we normally refer to as the y-coordinate) is negative, then it means we are looking at Quadrant IV.
Thus, we know that the complex number z lies in Quadrant IV.
If it lies in IV, then the angle that makes tangent -1 is;
theta=315° or 7pi/4 because that is the 45° or pi/4 angle in IV
Now, we have the modulus and theta so we can go ahead and substitute it into our trig form of z.
z=2sqrt2[cos(7pi/4) +isin(7pi/4)]
I hope this was well explained...
