$z=-2+8 i$ $\newline$ Find the angle $\theta$ (in radians) that $z$ makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express $\theta$ between $-\pi$ and $\pi$ . $\newline$ $\theta=$

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z=-2+8 i

\newline

Find the angle

\theta

(in radians) that

z

makes in the complex plane. Round your answer, if necessary, to the nearest thousandth. Express

\theta

between

-\pi

and

\pi

\newline

\theta=

Identify Real and Imaginary Parts: To find the angle $\theta$ that the complex number $z = -2 + 8i$ makes with the positive real axis in the complex plane, we need to calculate the argument of $z$ . The argument of a complex number is the angle formed by the radius (line from the origin to the point) and the positive x-axis. The argument can be found using the $\text{arctan}$ function, which is the inverse of the tangent function. The formula to find the argument of a complex number $a + bi$ is $\theta = \text{arctan}(b/a)$ , where $a$ is the real part and $b$ is the imaginary part of the complex number.
Calculate Argument using Arctan: First, we identify the real part $a$ and the imaginary part $b$ of the complex number $z = -2 + 8i$ . Here, $a = -2$ and $b = 8$ .
Determine Quadrant and Adjust Angle: Next, we calculate the argument $\theta$ using the arctan function: $\theta = \text{arctan}(b/a) = \text{arctan}(8/(-2)) = \text{arctan}(-4)$ . Since the complex number is in the second quadrant (because the real part is negative and the imaginary part is positive), the angle should be in the range $(\pi/2, \pi)$ .
Calculate Final Angle: Using a calculator, we find that $\arctan(-4) \approx -1.3258$ radians. However, since the angle is in the second quadrant, we need to add $\pi$ to this value to get the angle in the correct range. Therefore, $\theta = -1.3258 + \pi$ .
Verify Angle Range: Calculating the value of $\theta$ , we get $\theta \approx -1.3258 + 3.1416 \approx 1.8158$ radians.
Verify Angle Range: Calculating the value of $\theta$ , we get $\theta \approx -1.3258 + 3.1416 \approx 1.8158$ radians.We check to make sure that our final answer is within the range $-\pi$ to $\pi$ . Since $1.8158$ is between $-\pi$ and $\pi$ , our answer is in the correct range.