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प्रश्न
Express the following complex numbers in polar form and exponential form:
`1/(1 + "i")`
उत्तर
Let z = `1/(1 + "i")`
= `(1 - "i")/((1 + "i")(1 - "i"))`
= `(1 - "i")/(1 - "i"^2)`
= `(1 - "i")/(1 - (-1))` ...[∵ i2 = – 1]
= `(1 - "i")/2`
∴ z = `1/2 - 1/2"i"`
∴ a = `1/2`, b = `(-1)/2`
∴ | z | = r
= `sqrt("a"^2 + "b"^2)`
= `sqrt((1/2)^2 + (-1/2)^2)`
= `sqrt(1/4 + 1/4)`
= `1/sqrt(2)`
Here `(1/2, (-1)/2)` lies in 4th quadrant
θ = amp (z)
= `2pi + tan^-1("b"/"a")`
= `2pi + tan^-1(((-1)/2)/(1/2))`
= 2π + tan–1(–1)
= 2π – tan–1(1)
= `2pi - pi/4`
= `(7pi)/4`
∴ θ = 315° = `(7pi)/4`
∴ polar form of z = r (cos θ + i sin θ)
= `1/sqrt(2)(cos 315^circ + "i" sin315^circ)`
= `1/sqrt(2)[cos((7pi)/4) + "i" sin((7pi)/4)]`
The exponential form of z = reiθ
= `1/sqrt(2)"e"^((7pi)/4"i"`.
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