Advertisements
Advertisements
Question
Express the following complex numbers in polar form and exponential form:
`(1 + 2"i")/(1 - 3"i")`
Advertisements
Solution
Let z = `(1 + 2"i")/(1 - 3"i")`
= `(1 + 2"i")/(1 - 3"i") xx (1 + 3"i")/(1 + 3"i")`
= `(1 + 3"i" + 2"i" + 6"i"^2)/(1 - 9"i"^2)`
= `(1 + 5"i" - 6)/(1 + 9)` ...[∵ i2 = – 1]
∴ z = `(-5 + 5"i")/10`
= `-1/2 + 1/2`i
This is of the form a + bi, where a = `-1/2`, b = `1/2`
∴ r = `sqrt("a"^2 + "b"^2)`
= `sqrt((-1/2)^2 + (1/2)^2)`
= `sqrt(1/4 + 1/4)`
= `1/sqrt(2)`
Also, cos θ = `"a"/"r" = ((-1/2))/((1/sqrt(2))) = -1/sqrt(2)`
and sin θ = `"b"/"r" = ((1/2))/((1/sqrt(2))) = 1/sqrt(2)`
`∴ θ = (3pi)/4 ...[(because cos (3pi)/4 = cos (pi - pi/4) = -cos pi/4 = -1/sqrt(2)),(and sin (3pi)/4 = sin(pi - pi/4) = sin pi/4 = 1/sqrt(2))]`
∴ the polar form of z = r (cos θ + i sin θ)
= `1/sqrt(2)(cos (3pi)/4 + "i" sin (3pi)/4)`
and the exponential form of z = reiθ
= `1/sqrt(2)"e"^("i"((3pi)/4)`
= `1/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.
`sqrt(3) - "i"`
Find the modulus and amplitude of the following complex numbers.
3
Find the modulus and amplitude of the following complex numbers.
1 + i
Find the modulus and amplitude of the following complex numbers.
(1 + 2i)2 (1 − i)
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 complex numbers in polar form and exponential form:
−1
Express the following complex numbers in polar form and exponential form:
`1/(1 + "i")`
Express the following numbers in the form x + iy:
`sqrt(3)(cos pi/6 + "i" sin pi/6)`
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"^((-4pi)/3"i")`
Find the modulus and argument of the complex number `(1 + 2"i")/(1 - 3"i")`
Convert the complex number z = `("i" - 1)/(cos pi/3 + "i" sin pi/3)` in the polar form
For z = 2 + 3i verify the following:
`("z" + bar"z")` is real
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:
The modulus and argument of `(1 + "i"sqrt(3))^8` are respectively
Select the correct answer from the given alternatives:
If arg(z) = θ, then arg `bar(("z"))` =
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.
6 − i
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.
− 3i
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"`
The modulus of z = `sqrt7` + 3i is ______
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 A, B, C are three points in argand plane representing the complex numbers z1, z2 and z3 such that, z1 = `(λz_2 + z_3)/(λ + 1)`, where λ ∈ R, then find the distance of point A from the line joining points B and C.
