Advertisements
Advertisements
प्रश्न
Answer the following:
Convert the complex numbers in polar form and also in exponential form.
z = `-6 + sqrt(2)"i"`
Advertisements
उत्तर
z = `-6 + sqrt(2)"i"`
∴ a = – 6, b = `sqrt(2)`, i.e. a < 0, b > 0
∴ r = `sqrt("a"^2 + "b"^2)`
= `sqrt((-6)^2 + (sqrt(2))^2`
= `sqrt(36 + 2)`
= `sqrt(38)`
Here `(-6, sqrt(2))` lies in 2nd quadrant
∴ amp (z) = θ
= `pi + tan^-1("b"/"a")`
= `tan^-1(-sqrt(2)/6) + pi`
∴ the polar form of z = r(cos θ + i sin θ)
∴ `sqrt(38)(cos theta + "i" sin theta)`, where θ
= `pi + tan^-1(-sqrt(2)/6)`
∴ The exponential form of z = reiθ
`sqrt(38)"e" ^(pi + tan^-1(-sqrt(2)/6)`
APPEARS IN
संबंधित प्रश्न
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.
−3(1 − i)
Find the modulus and amplitude of the following complex number.
−4 − 4i
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)
Find real values of θ for which `((4 + 3"i" sintheta)/(1 - 2"i" sin theta))` is purely real.
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:
−1
Express the following complex numbers in polar form and exponential form:
`(1 + 2"i")/(1 - 3"i")`
Express the following complex numbers in polar form and exponential form:
`(1 + 7"i")/(2 - "i")^2`
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:
The modulus and argument of `(1 + "i"sqrt(3))^8` are respectively
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.
`(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:
Represent 1 + 2i, 2 − i, −3 − 2i, −2 + 3i by points in Argand's diagram.
The modulus of z = `sqrt7` + 3i is ______
The modulus and amplitude of 4 + 3i are ______
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.
