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Consider the complex number

z=6(cos(165^(@))+i sin(165^(@)))". "
Which of the following complex numbers best approximates
z ?
Choose 1 answer:
(A)
-5.8-1.6 i
(B)
-1.6+5.8 i
(C)
-5.8+1.6 i
(D)
-1.6-5.8 i

Consider the complex number $\newline$ $z=6\left(\cos \left(165^{\circ}\right)+i \sin \left(165^{\circ}\right)\right) \text {. }$ $\newline$ Which of the following complex numbers best approximates $z$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $-5.8-1.6 i$ $\newline$ (B) $-1.6+5.8 i$ $\newline$ (C) $-5.8+1.6 i$ $\newline$ (D) $-1.6-5.8 i$

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Compare linear, exponential, and quadratic growth

Full solution

Q. Consider the complex number

\newline

z=6\left(\cos \left(165^{\circ}\right)+i \sin \left(165^{\circ}\right)\right) \text {. }

\newline

Which of the following complex numbers best approximates

z

?

\newline

Choose

1

answer:

\newline

(A)

-5.8-1.6 i

\newline

(B)

-1.6+5.8 i

\newline

(C)

-5.8+1.6 i

\newline

(D)

-1.6-5.8 i

Convert to rectangular form: Convert the given complex number in polar form to rectangular form using the cosine and sine values. $\newline$ The complex number $z$ is given in polar form as $z = 6(\cos(165°) + i \sin(165°))$ . To convert it to rectangular form, we use the definitions of cosine and sine for the given angle.
Calculate cosine and sine: Calculate the cosine and sine of $165°$ . Using the unit circle or trigonometric tables, we find that: $\cos(165°) \approx -0.9659$ (since $165°$ is in the second quadrant where cosine is negative) $\sin(165°) \approx 0.2588$ (since $165°$ is in the second quadrant where sine is positive)
Multiply by modulus: Multiply the cosine and sine values by the modulus of the complex number. $\newline$ The modulus of $z$ is $6$ . Therefore, we multiply the cosine and sine values by $6$ to get the rectangular form: $\newline$ Real part: $6 \times \cos(165°) \approx 6 \times (-0.9659) \approx -5.7954$ $\newline$ Imaginary part: $6 \times \sin(165°) \approx 6 \times 0.2588 \approx 1.5528$
Round real and imaginary parts: Round the real and imaginary parts to one decimal place to match the answer choices. $\newline$ Real part rounded: $-5.8$ $\newline$ Imaginary part rounded: $1.6$
Form complex number: Form the complex number with the rounded real and imaginary parts. $\newline$ The approximate complex number in rectangular form is $-5.8 + 1.6i$ .
Match with answer choices: Match the approximate complex number with the given answer choices. $\newline$ The approximate complex number $-5.8 + 1.6i$ matches with choice (C) $-5.8 + 1.6i$ .

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