There we go with the first question on complex number ππ½ππ½
|Mathematics
There we go with the first question on complex number ππ½ππ½
ScanSolve AI Answer
Answer
in the range (βΟ,Ο]
Here's a step-by-step solution to the complex number problem:
Part a) Express z1β and z2β in the form r(cosΞΈ+isinΞΈ)
For a complex number z=x+yi, its polar form is z=r(cosΞΈ+isinΞΈ), where r=x2+y2β and ΞΈ=arctan(xyβ) adjusted for the correct quadrant, such that βΟ<ΞΈβ€Ο.
For z1β=1+3βi:
Step 1: Identify the real and imaginary parts.
x1β=1, y1β=3β.
Step 2: Calculate the modulus r1β.
r1β=x12β+y12ββ=12+(3β)2β=1+3β=4β=2
Step 3: Calculate the argument ΞΈ1β.
Since x1β>0 and y1β>0, ΞΈ1β is in the first quadrant.
tanΞΈ1β=x1βy1ββ=13ββ=3βΞΈ1β=arctan(3β)=3Οβ
This value is in the range (βΟ,Ο].
Step 4: Write z1β in polar form.
z1β=2(cos3Οβ+isin3Οβ)
For z2β=β1βi:
Step 1: Identify the real and imaginary parts.
x2β=β1, y2β=β1.
Step 2: Calculate the modulus r2β.
r2β=x22β+y22ββ=(β1)2+(β1)2β=1+1β=2β
Step 3: Calculate the argument ΞΈ2β.
Since x2β<0 and y2β<0, ΞΈ2β is in the third quadrant.
tanΞΈ2β=x2βy2ββ=β1β1β=1
The reference angle is 4Οβ. For the third quadrant, ΞΈ2β=β(Οβ4Οβ)=β43Οβ.
This value is in the range (βΟ,Ο].
Step 4: Write z2β in polar form.
z2β=2β(cos(β43Οβ)+isin(β43Οβ))
Part b) Evaluate βz2βz1βββ3 and arg(z2βz1ββ)3
Step 1: Find the polar form of z2βz1ββ.
If z1β=r1β(cosΞΈ1β+isinΞΈ1β) and z2β=r2β(cosΞΈ2β+isinΞΈ2β), then
z2βz1ββ=r2βr1ββ(cos(ΞΈ1ββΞΈ2β)+isin(ΞΈ1ββΞΈ2β))
Substitute the values from part a):
r2βr1ββ=2β2β=2βΞΈ1ββΞΈ2β=3Οββ(β43Οβ)=3Οβ+43Οβ=124Ο+9Οβ=1213Οβ
So,
z2βz1ββ=2β(cos(1213Οβ)+isin(1213Οβ))
Step 2: Evaluate βz2βz1βββ3.
Using De Moivre's Theorem, if Z=r(cosΞΈ+isinΞΈ), then β£Znβ£=rn.
Here, Z=z2βz1ββ, so r=2β and n=3.
βz2βz1βββ3=(2β)3=22β
Step 3: Evaluate arg(z2βz1ββ)3.
Using De Moivre's Theorem, arg(Zn)=nΞΈ.
Here, ΞΈ=1213Οβ and n=3.
arg(z2βz1ββ)3=3Γ1213Οβ=413Οβ
The argument must be in the range (βΟ,Ο]. We adjust 413Οβ by subtracting multiples of 2Ο:
413Οββ2Ο=413Οβ8Οβ=45Οβ
Since 45Οβ>Ο, we subtract 2Ο again:
45Οββ2Ο=45Οβ8Οβ=β43Οβ
This value is in the range $(-\pi, \pi
Was this helpful?
Need help with your own homework?
Get instant step-by-step solutions to any question. Free to start.
Here's a step-by-step solution to the complex number problem: Part a) Express z_1 and z_2 in the form r( + i) For a complex number z = x + yi, its polar form is z = r( + i), where r = sqrt(x^2 + y^2) and = ((y)/(x)) adjusted for the correct quadrant,β¦
There we go with the first question on complex number ππ½ππ½
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's a step-by-step solution to the complex number problem: Part a) Express z_1 and z_2 in the form r( + i) For a complex number z = x + yi, its polar form is z = r( + i), where r = sqrt(x^2 + y^2) and = ((y)/(x)) adjusted for the correct quadrant, such that - < . For z_1 = 1 + sqrt(3)i: Step 1: Identify the real and imaginary parts. x_1 = 1, y_1 = sqrt(3). Step 2: Calculate the modulus r_1. r_1 = sqrt(x_1^2 + y_1^2) = sqrt(1^2 + (3))^2 = sqrt(1 + 3) = sqrt(4) = 2 Step 3: Calculate the argument _1. Since x_1 > 0 and y_1 > 0, _1 is in the first quadrant. _1 = (y_1)/(x_1) = sqrt(3)1 = sqrt(3) _1 = (sqrt(3)) = ()/(3) This value is in the range (-, ]. Step 4: Write z_1 in polar form. z_1 = 2(()/(3) + i()/(3)) For z_2 = -1 - i: Step 1: Identify the real and imaginary parts. x_2 = -1, y_2 = -1. Step 2: Calculate the modulus r_2. r_2 = sqrt(x_2^2 + y_2^2) = sqrt((-1)^2 + (-1)^2) = sqrt(1 + 1) = sqrt(2) Step 3: Calculate the argument _2. Since x_2 < 0 and y_2 < 0, _2 is in the third quadrant. _2 = (y_2)/(x_2) = (-1)/(-1) = 1 The reference angle is ()/(4). For the third quadrant, _2 = -( - ()/(4)) = -(3)/(4). This value is in the range (-, ]. Step 4: Write z_2 in polar form. z_2 = sqrt(2)((-(3)/(4)) + i(-(3)/(4))) Part b) Evaluate |(z_1)/(z_2)|^3 and ((z_1)/(z_2))^3 Step 1: Find the polar form of (z_1)/(z_2). If z_1 = r_1(_1 + i_1) and z_2 = r_2(_2 + i_2), then (z_1)/(z_2) = (r_1)/(r_2)((_1 - _2) + i(_1 - _2)) Substitute the values from part a): (r_1)/(r_2) = (2)/(sqrt(2)) = sqrt(2) _1 - _2 = ()/(3) - (-(3)/(4)) = ()/(3) + (3)/(4) = (4 + 9)/(12) = (13)/(12) So, (z_1)/(z_2) = sqrt(2)(((13)/(12)) + i((13)/(12))) Step 2: Evaluate |(z_1)/(z_2)|^3. Using De Moivre's Theorem, if Z = r( + i), then |Z^n| = r^n. Here, Z = (z_1)/(z_2), so r = sqrt(2) and n=3. |(z_1)/(z_2)|^3 = (sqrt(2))^3 = 2sqrt(2) Step 3: Evaluate ((z_1)/(z_2))^3. Using De Moivre's Theorem, (Z^n) = n. Here, = (13)/(12) and n=3. ((z_1)/(z_2))^3 = 3 Γ (13)/(12) = (13)/(4) The argument must be in the range (-, ]. We adjust (13)/(4) by subtracting multiples of 2: (13)/(4) - 2 = (13 - 8)/(4) = (5)/(4) Since (5)/(4) > , we subtract 2 again: (5)/(4) - 2 = (5 - 8)/(4) = -(3)/(4) This value is in the range $(-,
