Advertisements
Advertisements
प्रश्न
Express the following complex numbers in polar form and exponential form:
`(1 + 2"i")/(1 - 3"i")`
उत्तर
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
संबंधित प्रश्न
Find the modulus and amplitude of the following complex numbers.
−3(1 − i)
Find the modulus and amplitude of the following complex numbers.
1 + i
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)
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/(1 + "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:
`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")` = |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)`
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 = .................
Select the correct answer from the given alternatives:
If z = x + iy and |z − zi| = 1 then
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.
− 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"`
Answer the following:
Represent 1 + 2i, 2 − i, −3 − 2i, −2 + 3i by points in Argand's diagram.
The polar coordinates of the point whose cartesian coordinates are (−2, −2), are given by ____________.
The modulus of z = `sqrt7` + 3i is ______
If x + iy = `5/(3 + costheta + isintheta)`, then x2 + y2 is equal to ______
If z = `5i ((-3)/5 i)`, then z is equal to 3 + bi. The value of ‘b’ 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 ______.
