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Consider the complex number z=-(5sqrt2)/(2)+(5sqrt2)/(2)i". " Which of the following complex numbers best approximates z^(3) ? Hint: z has a modulus of 5 and an argument of 135^(@). Choose 1 answer: (A) -10.6+10.6 i (B) -125 (c) -125 i (D) 88.4+88.4 i

Consider the complex number\newlinez=−522+522i  z=-\frac{5 \sqrt{2}}{2}+\frac{5 \sqrt{2}}{2} i \text { } \newlineWhich of the following complex numbers best approximates z3 z^{3} ? Hint: z z has a modulus of 55 and an argument of 135∘ 135^{\circ} .\newlineChoose 11 answer:\newline(A) −10.6+10.6i -10.6+10.6 i \newline(B) −125-125\newline(C) −125i -125 i \newline(D) 88.4+88.4i 88.4+88.4 i

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Q. Consider the complex number\newlinez=−522+522i  z=-\frac{5 \sqrt{2}}{2}+\frac{5 \sqrt{2}}{2} i \text { } \newlineWhich of the following complex numbers best approximates z3 z^{3} ? Hint: z z has a modulus of 55 and an argument of 135∘ 135^{\circ} .\newlineChoose 11 answer:\newline(A) −10.6+10.6i -10.6+10.6 i \newline(B) −125-125\newline(C) −125i -125 i \newline(D) 88.4+88.4i 88.4+88.4 i
  1. Identify modulus and argument: Identify the modulus and argument of the complex number zz. The complex number zz is given in the form z=a+biz = a + bi, where a=−522a = -\frac{5\sqrt{2}}{2} and b=522b = \frac{5\sqrt{2}}{2}. The modulus of zz is the distance from the origin to the point (a,b)(a, b) in the complex plane, which can be calculated using the formula ∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}.
  2. Calculate modulus of z: Calculate the modulus of z.\newline∣z∣=(−522)2+(522)2=(252)+(252)=25=5.|z| = \sqrt{\left(-\frac{5\sqrt{2}}{2}\right)^2 + \left(\frac{5\sqrt{2}}{2}\right)^2} = \sqrt{\left(\frac{25}{2}\right) + \left(\frac{25}{2}\right)} = \sqrt{25} = 5.\newlineThe modulus of z is indeed 55, as given in the hint.
  3. Determine argument of zz: Determine the argument of zz.\newlineThe argument of zz is the angle θ\theta made with the positive xx-axis. Given that zz has an argument of 135135 degrees, we can use this information directly without calculation.
  4. Express zz in polar form: Express zz in polar form.\newlineUsing the modulus and argument, we can express zz in polar form as z=∣z∣(cos⁡(θ)+isin⁡(θ))z = |z|(\cos(\theta) + i\sin(\theta)). For zz, this is z=5(cos⁡(135°)+isin⁡(135°))z = 5(\cos(135°) + i\sin(135°)).
  5. Calculate z3z^3 using De Moivre's Theorem: Calculate z(3)z^{(3)} using De Moivre's Theorem.\newlineDe Moivre's Theorem states that (r(cos⁡(θ)+isin⁡(θ)))n=rn(cos⁡(nθ)+isin⁡(nθ))(r(\cos(\theta) + i\sin(\theta)))^n = r^n(\cos(n\theta) + i\sin(n\theta)). For z(3)z^{(3)}, this becomes z(3)=53(cos⁡(3×135°)+isin⁡(3×135°))z^{(3)} = 5^3(\cos(3\times135°) + i\sin(3\times135°)).
  6. Compute value of z3z^3: Compute the value of z3z^{3}.z3=125(cos⁡(405°)+isin⁡(405°))z^{3} = 125(\cos(405°) + i\sin(405°)). Since 405°405° is equivalent to 45°45° (as we can subtract full rotations of 360°360°), we have z3=125(cos⁡(45°)+isin⁡(45°))z^{3} = 125(\cos(45°) + i\sin(45°)).
  7. Simplify expression for z3z^3: Simplify the expression for z3z^{3}.cos⁡(45°)=sin⁡(45°)=2/2\cos(45°) = \sin(45°) = \sqrt{2}/2. Therefore, z3=125(2/2+i2/2)z^{3} = 125(\sqrt{2}/2 + i\sqrt{2}/2).
  8. Calculate real and imaginary parts of z3z^3: Calculate the real and imaginary parts of z3z^{3}. The real part is 125(2/2)125(\sqrt{2}/2) and the imaginary part is 125(2/2)i125(\sqrt{2}/2)i. Multiplying these out gives us z3=1252/2+1252/2i≈88.4+88.4iz^{3} = 125\sqrt{2}/2 + 125\sqrt{2}/2i \approx 88.4 + 88.4i.

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