z=-3+5i 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=-3+5i 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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Calculate Argument: To find the angle theta that the complex number z = -3 + 5i makes with the positive real axis in the complex plane, we need to calculate the argument of the complex number. The argument is the angle in polar coordinates and can be found using the arctan function, which gives the angle whose tangent is the quotient of the imaginary part over the real part of the complex number.Find Tangent Ratio: The complex number z = -3 + 5i has a real part of -3 and an imaginary part of 5. The tangent of the angle theta is the ratio of the imaginary part to the real part, which is 5 / -3. Therefore, we use the arctan function to find the angle: theta = arctan(5 / -3).Adjust for Quadrant: However, since the real part of z is negative and the imaginary part is positive, z lies in the second quadrant of the complex plane. The range of the arctan function is (-pi/2, pi/2), but we need to adjust the angle because the arctan function does not account for the quadrant in which the complex number lies. To find the correct angle in the second quadrant, we add pi to the arctan value.Calculate Arctan Value: Calculating the arctan value: theta = arctan(5 / -3) = arctan(-5/3). Using a calculator, we find that arctan(-5/3) ≈ -1.0304 radians. Since this is in the fourth quadrant and we need the angle in the second quadrant, we add pi to get the correct angle: theta = -1.0304 + pi.Add Pi: Adding pi to -1.0304 radians, we get theta ≈ -1.0304 + 3.1416 = 2.1112 radians. Since we want the angle to be expressed between -pi and pi, and 2.1112 radians is within this range, this is our final answer.

$z=-3+5 i$ $\newline$ 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=−3+5i z=-3+5 i z=−3+5i\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=

$z=-3+5 i$ $\newline$ 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 $\pi$ . $\newline$ $\theta=$