Advertisements
Advertisements
प्रश्न
Write in polar form of the following complex numbers
– 2 – i2
उत्तर
– 2 – i2 = r(cos θ + i sin θ)
Let z = – 2 – i2 = r(cos θ + i sin θ)
Equating real and imaginary parts
r cos θ = – 2
r sin θ = – 2
r2 cos2θ + r2 sin2θ = (– 2)2 + (– 2)2
r2 = 4 + 4 = 8
r2 = 8
|z| = r = `2sqrt(2)`
cos θ = `(-2)/(2sqrt(2) = (-1)/sqrt(2)`
sin θ = `(-2)/(2sqrt(2)) = (-1)/sqrt(2)`
Since cos θ and sin θ both are in – ve so lies in III quadrant.
Argument = `2"k"pi - 3 pi/4`
As θ = `pi/4 - pi = - (3pi)/4`
∴ Polar from z = r(cos θ + i sin θ)
– 2 – i2 = `2sqrt(2) (cos(2"k"pi - (3pi)/4) + "i" sin(2"k"pi - 3/4)) "k" ∈ "z"`
APPEARS IN
संबंधित प्रश्न
Write in polar form of the following complex numbers
`3 - "i"sqrt(3)`
Write in polar form of the following complex numbers
`("i" - 1)/(cos pi/3 + "i" sin pi/3)`
Find the rectangular form of the complex numbers
`(cos pi/6 "i" sin pi/6)(cos pi/12 + "i" sin pi/12)`
Find the rectangular form of the complex numbers
`(cos pi/6 - "i" sin pi/6)/(2(cos pi/3 + "i" sin pi/3))`
If (x1 + iy1)(x2 + iy2)(x3 + iy3) ... (xn + iyn) = a + ib, show that `(x_1^2 + y_1^2)(x_2^2 + y_2^2)(x_3^2 + y_3^2) ... (x_"n"^2 + y_"n"^2)` = a2 + b2
If (x1 + iy1)(x2 + iy2)(x3 + iy3) ... (xn + iyn) = a + ib, show that `sum_("r" = 1)^"n" tan^-1 (y_"r"/x_"r") = tan^-1 ("b"/"a") + 2"k"pi, "k" ∈ "z"`
If `(1 + z)/(1 - z)` = cos 2θ + i sin 2θ, show that z = i tan θ
If cos α + cos β + cos γ = sin α + sin β + sin γ = 0. then show that sin 3α + sin 3β + sin 3γ = 3 sin(α + β + γ)
If z = x + iy and arg `((z - "i")/(z + 2)) = pi/4`, show that x2 + y3 + 3x – 3y + 2 = 0
Choose the correct alternative:
The solution of the equation |z| – z = 1 + 2i is
Choose the correct alternative:
If z is a complex number such that z ∈ C\R and `"z" + 1/"z"` ∈ R, then |z| is
Choose the correct alternative:
The principal argument of `3/(-1 + "i")` is
