Advertisements
Advertisements
Question
Convert the complex number z = `("i" - 1)/(cos pi/3 + "i" sin pi/3)` in the polar form
Solution
z = `("i" - 1)/(cos pi/3 + "i" sin pi/3)`
= `("i" - 1)/(1/2 + "i"(sqrt(3)/2))`
= `(2"i" - 2)/(1 + sqrt(3)"i")`
= `(2"i" - 2)/(1 + sqrt(3)"i") xx (1 - sqrt(3)"i")/(1 - sqrt(3)"i")`
= `(2"i" - 2sqrt(3)"i"^2 - 2 + 2sqrt(3)"i")/(1 - 3"i"^2)`
= `(2"i" + 2sqrt(3) - 2 + 2sqrt(3)"i")/(1 + 3)` ...[∵ i2 = – 1]
= `((-2 + 2sqrt(3)) + (2 + 2sqrt(3))"i")/4`
∴ z = `((-1 + sqrt(3))/2) + ((1 + sqrt(3))/2)"i"`
This is of the form a + bi, where
a = `(-1 + sqrt(3))/2` and b = `(1 + sqrt(3))/2`
∴ r = `sqrt("a"^2 + "b"^2)`
= `sqrt(((sqrt(3) - 1)/2)^2 + ((sqrt(3) + 1)/2)^2`
= `sqrt((3 + 1 - 2sqrt(3))/4 + (3 + 1 + 2sqrt(3))/4)`
= `sqrt(8/4)`
= `sqrt(2)`
Also, cos θ = `"a"/"r" = (sqrt(3) - 1)/(2sqrt(2))`
and sin θ = `"b"/"r" -= (sqrt(3) + 1)/(2sqrt(2))`
∴ tan θ =`(sqrt(3) + 1)/(sqrt(3) - 1)`
∴ the polar form of z = r(cos θ + i sin θ)
= `sqrt(2)(cos theta + "i" sin theta)`,
where tan θ = `(sqrt(3) + 1)/(sqrt(3) - 1)`
APPEARS IN
RELATED QUESTIONS
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.
−4 − 4i
Find the modulus and amplitude of the following complex numbers.
`1 + "i"sqrt(3)`
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:
− 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 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"^((-4pi)/3"i")`
Express the following numbers in the form x + iy:
`"e"^((5pi)/6"i")`
Find the modulus and argument of the complex number `(1 + 2"i")/(1 - 3"i")`
For z = 2 + 3i verify the following:
`"z"bar("z")` = |z|2
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)`
z1 = 1 + i, z2 = 2 − 3i. Verify the following :
`bar("z"_1."z"_2) = bar("z"_1).bar("z"_2)`
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 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.
`(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.
Answer the following:
Convert the complex numbers in polar form and also in exponential form.
z = `(2 + 6sqrt(3)"i")/(5 + sqrt(3)"i")`
The polar coordinates of the point whose cartesian coordinates are (−2, −2), are given by ____________.
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 ______.
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.
