1. `A=5+6i`
Find Exponential form of complex numberSolution:Here `A=5+6i`
For a complex number `z=a+bi`, the exponential form `r*e^(i*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 exponential form is `r*e^(i*theta)`
`=7.8102*e^(i(50.1944))` (in degree)
`=7.8102*e^(i(0.8761))` (in radian)
2. `A=-2+3i`
Find Exponential form of complex numberSolution:Here `A=-2+3i`
For a complex number `z=a+bi`, the exponential form `r*e^(i*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 exponential form is `r*e^(i*theta)`
`=3.6056*e^(i(123.6901))` (in degree)
`=3.6056*e^(i(2.1588))` (in radian)
3. `A=1-3i`
Find Exponential form of complex numberSolution:Here `A=1-3i`
For a complex number `z=a+bi`, the exponential form `r*e^(i*theta)`
where `r=sqrt(a^2+b^2)` and `theta=atan(b/a)`
Here, `a=1` and `b=-3`
`:. r=sqrt(1^2+(-3)^2)=sqrt(1+9)=sqrt(10)=3.1623`
`theta=atan(b/a)` (Since `a>0`)
`:. theta=atan((-3)/(1))`
`:. theta=atan(-3)`
`:. theta=-71.5651 ^circ` or `theta=-1.249` rad
The exponential form is `r*e^(i*theta)`
`=3.1623*e^(i(-71.5651))` (in degree)
`=3.1623*e^(i(-1.249))` (in radian)
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