z=-2+8i 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. theta=

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z=-2+8i 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.  theta=

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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 arctan function, which is the inverse of the tangent function. The formula to find the argument of a complex number a + bi is theta = 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 = arctan(b/a) = arctan(8/(-2)) = 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) ≈ -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 ≈ -1.3258 + 3.1416 ≈ 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.

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

$z=-2+8 i$ $\newline$ 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 $\pi$ . $\newline$ $\theta=$