$z=-3-6 i$ $\newline$ 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=-3-6 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 $z = -3 - 6i$ makes with the positive x-axis in the complex plane, we need to use the arctangent function, which gives us the angle in radians for a given ratio of the imaginary part to the real part of a complex number. The formula to find the angle is $\theta = \text{atan2}(\text{imaginary part}, \text{real part})$ .
Use atan $2$ function: First, identify the real part and the imaginary part of the complex number $z = -3 - 6i$ . The real part is $-3$ and the imaginary part is $-6$ .
Calculate $\theta$ : Now, use the $\text{atan2}$ function to find the angle. The $\text{atan2}$ function takes into account the signs of both the real and imaginary parts to determine the correct quadrant for the angle. $\newline$ $\theta = \text{atan2}(-6, -3)$
Determine the quadrant: Calculate the value of $\theta$ using the atan $2$ function. $\theta = \text{atan2}(-6, -3)$
Find the angle $\theta$ : The $\text{atan2}$ function will return a value in radians. Since the real part and the imaginary part are both negative, the angle $\theta$ will be in the third quadrant. The $\text{atan2}$ function will automatically give us an angle in the correct range between $-\pi$ and $\pi$ .
Find the angle $\theta$ : The $\text{atan2}$ function will return a value in radians. Since the real part and the imaginary part are both negative, the angle $\theta$ will be in the third quadrant. The $\text{atan2}$ function will automatically give us an angle in the correct range between $-\pi$ and $\pi$ .After calculating, we find that the angle $\theta$ is approximately $-2.034$ radians. This is the angle that $z$ makes with the positive $x$ -axis in the complex plane, and it is expressed between $-\pi$ and $\pi$ .