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Consider the complex number

z=4(cos(120^(@))+i sin(120^(@)))". "
Which of the following complex numbers best approximates
z ?
Choose 1 answer:
(A)
3.5-2i
(B)
2-3.5 i
(C)
-2+3.5 i
(D)
-3.5+2i

Consider the complex number $\newline$ $z=4\left(\cos \left(120^{\circ}\right)+i \sin \left(120^{\circ}\right)\right) \text {. }$ $\newline$ Which of the following complex numbers best approximates $z$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $3.5-2 i$ $\newline$ (B) $2-3.5 i$ $\newline$ (C) $-2+3.5 i$ $\newline$ (D) $-3.5+2 i$

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Compare linear, exponential, and quadratic growth

Full solution

Q. Consider the complex number

\newline

z=4\left(\cos \left(120^{\circ}\right)+i \sin \left(120^{\circ}\right)\right) \text {. }

\newline

Which of the following complex numbers best approximates

z

?

\newline

Choose

1

answer:

\newline

(A)

3.5-2 i

\newline

(B)

2-3.5 i

\newline

(C)

-2+3.5 i

\newline

(D)

-3.5+2 i

Convert to standard form: Convert the given complex number in trigonometric form to standard form. $\newline$ We have $z = 4(\cos(120°) + i \sin(120°))$ . To convert this to standard form, we calculate the cosine and sine of $120°$ . $\newline$ Cosine and sine of $120°$ can be found using the unit circle or trigonometric tables: $\newline$ $\cos(120°) = -\frac{1}{2}$ $\newline$ $\sin(120°) = \frac{\sqrt{3}}{2}$ $\newline$ Now, we multiply these values by $4$ (the modulus of the complex number): $\newline$ Real part: $4 \cdot \cos(120°) = 4 \cdot (-\frac{1}{2}) = -2$ $\newline$ Imaginary part: $4 \cdot i \cdot \sin(120°) = 4 \cdot i \cdot (\frac{\sqrt{3}}{2}) = 2\sqrt{3} \cdot i$
Calculate cosine and sine: Approximate the value of $\sqrt{3}$ to compare with the given options. $\newline$ The value of $\sqrt{3}$ is approximately $1.732$ . So, the imaginary part of the complex number is approximately: $\newline$ $2\sqrt{3} \cdot i \approx 2 \cdot 1.732 \cdot i \approx 3.464 \cdot i$ $\newline$ We can round this to $3.5 \cdot i$ for comparison purposes.
Multiply by modulus: Compare the standard form of the complex number with the given options. $\newline$ The standard form of the complex number is approximately $-2 + 3.5i$ . Now we compare this with the given options: $\newline$ (A) $3.5 - 2i$ $\newline$ (B) $2 - 3.5i$ $\newline$ (C) $-2 + 3.5i$ $\newline$ (D) $-3.5 + 2i$ $\newline$ Option (C) matches our calculated standard form of the complex number.

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