1. `A=-1-i,B=-2+3i,C=1-3i`
Find polar(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))`
where `r=sqrt(a^2+b^2)` and `theta=atan(b/a)`
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
`:.` The polar form (in degree) is `1.4142*(cos(225)+i*sin(225))`
`:.` The polar form (in radian) is `1.4142*(cos((5pi)/(4))+i*sin((5pi)/(4)))`
2. `A=5+6i,B=-2+3i,C=1-3i`
Find polar(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))`
where `r=sqrt(a^2+b^2)` and `theta=atan(b/a)`
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
`:.` The polar form (in degree) is `7.8102*(cos(50.1944)+i*sin(50.1944))`
`:.` The polar form (in radian) is `7.8102*(cos(0.8761)+i*sin(0.8761))`
3. `A=-1-i,B=-2+3i,C=1-3i`
Find polar(B)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))`
where `r=sqrt(a^2+b^2)` and `theta=atan(b/a)`
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
`:.` The polar form (in degree) is `3.6056*(cos(123.6901)+i*sin(123.6901))`
`:.` The polar form (in radian) is `3.6056*(cos(2.1588)+i*sin(2.1588))`
This material is intended as a summary. Use your textbook for detail explanation.
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