Consider the complex number $z=5\left(\cos \left(15^{\circ}\right)+i \sin \left(15^{\circ}\right)\right)$ . $\newline$ Which of the following complex numbers best approximates $z$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $4.8+1.3 i$ $\newline$ (B) $1.3+4.8 i$ $\newline$ (C) $-1.3+4.8 i$ $\newline$ (D) $-4.8+1.3 i$

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Algebra 1

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Q. Consider the complex number

z=5\left(\cos \left(15^{\circ}\right)+i \sin \left(15^{\circ}\right)\right)

\newline

Which of the following complex numbers best approximates

z

\newline

Choose

1

answer:

\newline

(A)

4.8+1.3 i

\newline

(B)

1.3+4.8 i

\newline

(C)

-1.3+4.8 i

\newline

(D)

-4.8+1.3 i

Given complex number $z$ in polar form: We are given the complex number $z$ in polar form: $\newline$ $z = 5(\cos(15°) + i \sin(15°))$ . $\newline$ To find the complex number that best approximates $z$ , we need to convert this polar form into its rectangular form, which is $z = a + bi$ , where $a$ is the real part and $b$ is the imaginary part.
Calculating the real part of z: First, we calculate the real part of $z$ , which is $a = 5 \times \cos(15°)$ . Using a calculator or trigonometric tables, we find that $\cos(15°)$ is approximately $0.9659$ . So, $a \approx 5 \times 0.9659$ .
Calculating the imaginary part of z: Now, we perform the calculation for the real part: $\newline$ $a \approx 5 \times 0.9659 \approx 4.8295$ . $\newline$ We can round this to one decimal place, getting $a \approx 4.8$ .
Converting to rectangular form: Next, we calculate the imaginary part of $z$ , which is $b = 5 \times \sin(15°)$ . Using a calculator or trigonometric tables, we find that $\sin(15°)$ is approximately $0.2588$ . So, $b \approx 5 \times 0.2588$ .
Comparing the approximation to options: Now, we perform the calculation for the imaginary part: $\newline$ $b \approx 5 \times 0.2588 \approx 1.294$ . $\newline$ We can round this to one decimal place, getting $b \approx 1.3$ .
Comparing the approximation to options: Now, we perform the calculation for the imaginary part: $\newline$ $b \approx 5 \times 0.2588 \approx 1.294$ . $\newline$ We can round this to one decimal place, getting $b \approx 1.3$ .Combining the real and imaginary parts, we get the rectangular form of $z$ : $\newline$ $z \approx 4.8 + 1.3i$ .
Comparing the approximation to options: Now, we perform the calculation for the imaginary part: $\newline$ $b \approx 5 \times 0.2588 \approx 1.294$ . $\newline$ We can round this to one decimal place, getting $b \approx 1.3$ .Combining the real and imaginary parts, we get the rectangular form of $z$ : $\newline$ $z \approx 4.8 + 1.3i$ .We compare this approximation to the given options: $\newline$ (A) $4.8 + 1.3i$ $\newline$ (B) $1.3 + 4.8i$ $\newline$ (C) $-1.3 + 4.8i$ $\newline$ (D) $-4.8 + 1.3i$ $\newline$ The approximation we calculated, $4.8 + 1.3i$ , matches option (A).