A complex number z is of the form
z=a+bi
We define the Polar coordinates of z to be (r,theta), as seen in the image below.
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From this diagram we get some more properties:
r=sqrt(a^2+b^2)
sin theta = b/r => b=rsintheta
costheta=a/r =>a=rcostheta
If we substitute b and a into the definition of a complex number, we have
z=rcostheta+irsintheta = r(costheta+isintheta)
Our trigonometric sum resembles color(red)("Euler's identity"):
e^(icolor(red)alpha)=coscolor(red)alpha+isincolor(Red)(alpha
Thus,
z=re^(itheta)
In our case, it'd be easier to write them in this exponential form then transform it into trigonometric form.
Let color(blue)(z_1 = -i+1 and color(blue)(z_2 = 2i+10.
We do not need to find theta_1 and theta_2 now, so we will let them as that.
r_1 = sqrt(a_1^2+b_1^2) = sqrt(1^2+(-1)^2) = sqrt2
r_2 = sqrt(a_2^2+b_2^2)=sqrt(100+4)=sqrt(104)=2sqrt(26)
:. z_1/z_2=(r_1e^(itheta_1))/(r_2e^(itheta_2))
z_1/z_2 = sqrt2/(2sqrt26) * e^(i(theta_1-theta_2)
z_1/z_2 = 1/(2sqrt13) * e^(i(theta_1-theta_2)
We can still apply Euler's identity to e^(i(theta_1-theta_2)). We have:
e^(i(theta_1-theta_2)) = cos(theta_1-theta_2)+isin(theta_1-theta_2)
The color(blue)("Difference formula") for cosine and sine is, as follows:
cos(a-b)=cosacosb+sinasinb
sin(a-b) = sinacosb-cosasinb
cos(theta_1 - theta_2) = costheta_1costheta_2+sintheta_1sintheta_2
sin(theta_1-theta_2) = sintheta_1costheta_2-costheta_1sintheta_2
From the properties we got earlier, we know:
costheta_1 = 1/sqrt2
sintheta_1 = -1/sqrt2
costheta_2 = 5/sqrt26
sintheta_2=1/sqrt26
After we calculate the values we needed, we reach this:
cos(theta_1-theta_2) = 2/sqrt13
sin(theta_1-theta_2)= -3/sqrt13
Finally, we get:
z_1/z_2 = 1/(2sqrt13) (2/sqrt13 -i3/sqrt13)
z_1/z_2 = 1/2(2/13 - i3/13)
:.
(-i+1)/(2i+10) =1/2(2/13 - i3/13)
