Advertisements
Advertisements
Question
Express the following complex numbers in polar form and exponential form:
`(1 + 7"i")/(2 - "i")^2`
Advertisements
Solution
Let z = `(1 + 7"i")/(2 - "i")^2`
= `(1 + 7"i")/(4 - 4"i" + "i"^2)`
= `(1 + 7"i")/(4 - 4"i" - 1)`
= `(1 + 7"i")/(3 - 4"i")`
= `((1 + 7"i")(3 + 4"i"))/((3 - 4"i")(3 + 4"i"))`
= `(3 + 4"i" + 21"i" + 28"i"^2)/(9 - 16"i"^2)`
= `(25"i" + 3 + 28(-1))/(9 - 16(-1))`
= `(25"i" - 25)/25`
∴ z = – 1 + i
∴ a = – 1, b = 1
∴ |z| = r
= `sqrt("a"^2 + "b"^2)`
= `sqrt((-1)^2 + 1^2)`
= `sqrt(2)`
Here, (–1, 1) lies in 2nd quadrant
θ = amp (z)
= `pi + tan^-1("b"/"a")`
= `pi + tan^-1(1/(-1))`
= π + tan–1(–1)
= π – tan–1(1)
= `pi - pi/4`
= `(3pi)/4`
∴ θ = 135° = `(3pi)/4`
∴ the polar form of z = r (cos θ + i sin θ)
= `sqrt(2) (cos 135^circ + "i" sin 135^circ)`
= `sqrt(2)(cos (3pi)/4 + "i" sin (3pi)/4)`
The exponential form of z = reiθ = `sqrt(2)"e"^((3pi)/4"i"`
APPEARS IN
RELATED QUESTIONS
Find the modulus and amplitude of the following complex numbers.
7 − 5i
Find the modulus and amplitude of the following complex numbers.
`sqrt(3) + sqrt(2)"i"`
Find the modulus and amplitude of the following complex numbers.
−8 + 15i
Find the modulus and amplitude of the following complex numbers.
−3(1 − i)
Find the modulus and amplitude of the following complex numbers.
3
Find the modulus and amplitude of the following complex numbers.
1 + i
If z = 3 + 5i then represent the `"z", bar("z"), - "z", bar(-"z")` in Argand's diagram
Express the following complex numbers in polar form and exponential form:
`-1 + sqrt(3)"i"`
Express the following complex numbers in polar form and exponential form:
− i
Express the following numbers in the form x + iy:
`sqrt(2)(cos (7pi)/4 + "i" sin (7pi)/4)`
Express the following numbers in the form x + iy:
`7(cos(-(5pi)/6) + "i" sin (- (5pi)/6))`
Express the following numbers in the form x + iy:
`"e"^(pi/3"i")`
Express the following numbers in the form x + iy:
`"e"^((5pi)/6"i")`
For z = 2 + 3i verify the following:
`bar((bar"z"))` = z
For z = 2 + 3i verify the following:
`"z" - bar"z"` = 6i
z1 = 1 + i, z2 = 2 − 3i. Verify the following :
`bar("z"_1."z"_2) = bar("z"_1).bar("z"_2)`
Select the correct answer from the given alternatives:
If `-1 + sqrt(3)"i"` = reiθ , then θ = .................
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
8 + 15i
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
6 − i
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
`(1 + sqrt(3)"i")/2`
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
`(-1 - "i")/sqrt(2)`
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
2i
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
`1/sqrt(2) + 1/sqrt(2)"i"`
Answer the following:
Represent 1 + 2i, 2 − i, −3 − 2i, −2 + 3i by points in Argand's diagram.
Answer the following:
Convert the complex numbers in polar form and also in exponential form.
z = `(2 + 6sqrt(3)"i")/(5 + sqrt(3)"i")`
Convert the complex numbers in polar form and also in exponential form.
`(-3)/2 + (3sqrt(3))/2"i"`
The modulus and amplitude of 4 + 3i are ______
If x + iy = `5/(3 + costheta + isintheta)`, then x2 + y2 is equal to ______
For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 - 3 - 4i| = 5, the minimum value of |z1 - z2| is ______.
If z = `π/4(1 + i)^4((1 - sqrt(π)i)/(sqrt(π) + i) + (sqrt(π) - i)/(1 + sqrt(π)i))`, then `(|z|/("amp"^((z))))` is equals to ______.
