1. `A=-1-i,B=-2+3i,C=1-3i`
Find roots(4,A)Solution:Here `A=-1-i,B=-2+3i,C=1-3i`
For a complex number `z=a+bi`, the polar form is `z=r*(cos(theta)+i*sin(theta))`
then all `n^(th)` roots of given complex number can be obtained by the De Moivre's Formula i.e.
`w_k=[r*(cos(theta)+i*sin(theta))]^(1/n)=root (n)(r)*[cos((theta+2kpi)/n)+i*sin((theta+2kpi)/n)]`, where `k=0,1,2..,n-1`
Here `n=4`
Step-1: Convert to polar form: `z=r*(cos(theta)+i*sin(theta))`Here, `a=-1` and `b=-1`
`:. r=sqrt((-1)^2+(-1)^2)=sqrt(1+1)=sqrt(2)=1.4142`
`theta=atan(b/a)+180` (Since `a<0`)
`:. theta=atan((-1)/(-1))+180`
`:. theta=atan(1)+180`
`:. theta=45+180`
`:. theta=225 ^circ` or `theta=(5pi)/(4)` rad = 3.927 rad
`:. theta=3.927`
The polar form is `r*(cos(theta)+i*sin(theta))`
`=1.4142*(cos(3.927)+i*sin(3.927))`
Step-2: All `n^(th)` roots of given complex number `r*(cos(theta)+i*sin(theta))` is
`w_k=root (n)(r)*[cos((theta+2kpi)/n)+i*sin((theta+2kpi)/n)]`, `k=0,..,n-1`Hence the 4 roots are
for `k=0``w_0=1.0905*[cos((3.927+2*0*pi)/4)+i*sin((3.927+2*0*pi)/4)]`
`w_0=1.0905*[cos(0.9817)+i*sin(0.9817)]`
`w_0=0.6059+0.9067i`
for `k=1``w_1=1.0905*[cos((3.927+2*1*pi)/4)+i*sin((3.927+2*1*pi)/4)]`
`w_1=1.0905*[cos(2.5525)+i*sin(2.5525)]`
`w_1=-0.9067+0.6059i`
for `k=2``w_2=1.0905*[cos((3.927+2*2*pi)/4)+i*sin((3.927+2*2*pi)/4)]`
`w_2=1.0905*[cos(4.1233)+i*sin(4.1233)]`
`w_2=-0.6059-0.9067i`
for `k=3``w_3=1.0905*[cos((3.927+2*3*pi)/4)+i*sin((3.927+2*3*pi)/4)]`
`w_3=1.0905*[cos(5.6941)+i*sin(5.6941)]`
`w_3=0.9067-0.6059i`
2. `A=5+6i,B=-2+3i,C=1-3i`
Find roots(4,A)Solution:Here `A=5+6i,B=-2+3i,C=1-3i`
For a complex number `z=a+bi`, the polar form is `z=r*(cos(theta)+i*sin(theta))`
then all `n^(th)` roots of given complex number can be obtained by the De Moivre's Formula i.e.
`w_k=[r*(cos(theta)+i*sin(theta))]^(1/n)=root (n)(r)*[cos((theta+2kpi)/n)+i*sin((theta+2kpi)/n)]`, where `k=0,1,2..,n-1`
Here `n=4`
Step-1: Convert to polar form: `z=r*(cos(theta)+i*sin(theta))`Here, `a=5` and `b=6`
`:. r=sqrt(5^2+6^2)=sqrt(25+36)=sqrt(61)=7.8102`
`theta=atan(b/a)` (Since `a>0`)
`:. theta=atan((6)/(5))`
`:. theta=atan(1.2)`
`:. theta=50.1944 ^circ` or `theta=0.8761` rad
`:. theta=0.8761`
The polar form is `r*(cos(theta)+i*sin(theta))`
`=7.8102*(cos(0.8761)+i*sin(0.8761))`
Step-2: All `n^(th)` roots of given complex number `r*(cos(theta)+i*sin(theta))` is
`w_k=root (n)(r)*[cos((theta+2kpi)/n)+i*sin((theta+2kpi)/n)]`, `k=0,..,n-1`Hence the 4 roots are
for `k=0``w_0=1.6717*[cos((0.8761+2*0*pi)/4)+i*sin((0.8761+2*0*pi)/4)]`
`w_0=1.6717*[cos(0.219)+i*sin(0.219)]`
`w_0=1.6318+0.3632i`
for `k=1``w_1=1.6717*[cos((0.8761+2*1*pi)/4)+i*sin((0.8761+2*1*pi)/4)]`
`w_1=1.6717*[cos(1.7898)+i*sin(1.7898)]`
`w_1=-0.3632+1.6318i`
for `k=2``w_2=1.6717*[cos((0.8761+2*2*pi)/4)+i*sin((0.8761+2*2*pi)/4)]`
`w_2=1.6717*[cos(3.3606)+i*sin(3.3606)]`
`w_2=-1.6318-0.3632i`
for `k=3``w_3=1.6717*[cos((0.8761+2*3*pi)/4)+i*sin((0.8761+2*3*pi)/4)]`
`w_3=1.6717*[cos(4.9314)+i*sin(4.9314)]`
`w_3=0.3632-1.6318i`
3. `A=5+6i,B=-2+3i,C=1-3i`
Find roots(3,B)Solution:Here `A=5+6i,B=-2+3i,C=1-3i`
For a complex number `z=a+bi`, the polar form is `z=r*(cos(theta)+i*sin(theta))`
then all `n^(th)` roots of given complex number can be obtained by the De Moivre's Formula i.e.
`w_k=[r*(cos(theta)+i*sin(theta))]^(1/n)=root (n)(r)*[cos((theta+2kpi)/n)+i*sin((theta+2kpi)/n)]`, where `k=0,1,2..,n-1`
Here `n=3`
Step-1: Convert to polar form: `z=r*(cos(theta)+i*sin(theta))`Here, `a=-2` and `b=3`
`:. r=sqrt((-2)^2+3^2)=sqrt(4+9)=sqrt(13)=3.6056`
`theta=atan(b/a)+180` (Since `a<0`)
`:. theta=atan((3)/(-2))+180`
`:. theta=atan(-1.5)+180`
`:. theta=-56.3099+180`
`:. theta=123.6901 ^circ` or `theta=2.1588` rad
`:. theta=2.1588`
The polar form is `r*(cos(theta)+i*sin(theta))`
`=3.6056*(cos(2.1588)+i*sin(2.1588))`
Step-2: All `n^(th)` roots of given complex number `r*(cos(theta)+i*sin(theta))` is
`w_k=root (n)(r)*[cos((theta+2kpi)/n)+i*sin((theta+2kpi)/n)]`, `k=0,..,n-1`Hence the 3 roots are
for `k=0``w_0=1.5334*[cos((2.1588+2*0*pi)/3)+i*sin((2.1588+2*0*pi)/3)]`
`w_0=1.5334*[cos(0.7196)+i*sin(0.7196)]`
`w_0=1.1532+1.0106i`
for `k=1``w_1=1.5334*[cos((2.1588+2*1*pi)/3)+i*sin((2.1588+2*1*pi)/3)]`
`w_1=1.5334*[cos(2.814)+i*sin(2.814)]`
`w_1=-1.4519+0.4934i`
for `k=2``w_2=1.5334*[cos((2.1588+2*2*pi)/3)+i*sin((2.1588+2*2*pi)/3)]`
`w_2=1.5334*[cos(4.9084)+i*sin(4.9084)]`
`w_2=0.2986-1.504i`
This material is intended as a summary. Use your textbook for detail explanation.
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