$z=-7-6 i$ $\newline$ Find the angle $\theta$ (in degrees) that $z$ makes in the complex plane. $\newline$ Round your answer, if necessary, to the nearest tenth. Express $\theta$ between $-180^{\circ}$ and $180^{\circ}$ . $\newline$ $\theta=\square^{\circ}$

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Full solution

z=-7-6 i

\newline

Find the angle

\theta

(in degrees) that

z

makes in the complex plane.

\newline

Round your answer, if necessary, to the nearest tenth. Express

\theta

between

-180^{\circ}

and

180^{\circ}

\newline

\theta=\square^{\circ}

Understand the problem: Understand the problem. $\newline$ We need to find the angle $\theta$ that the complex number $z = -7 - 6i$ makes with the positive x-axis in the complex plane. This angle is also known as the argument of the complex number.
Represent in complex plane: Represent the complex number in the complex plane. $\newline$ The complex number $z = -7 - 6i$ can be represented as a point in the complex plane with coordinates $(-7, -6)$ . The angle $\theta$ we are looking for is the angle between the positive x-axis and the line segment from the origin to the point $(-7, -6)$ , measured in the counter-clockwise direction.
Calculate using arctangent: Calculate the angle using the arctangent function. $\newline$ The angle $\theta$ can be found using the arctangent of the imaginary part divided by the real part of the complex number. However, since the complex number is in the third quadrant, we need to add $180^\circ$ to the result of the arctangent to get the correct angle in the range of $-180^\circ$ to $180^\circ$ . $\newline$ $\theta = \text{atan2}(-6, -7) + 180^\circ$
Use calculator to find angle: Use a calculator to find the angle. $\newline$ Using a calculator with the atan $2$ function, we input the imaginary part $-6$ and the real part $-7$ to get the angle in radians. Then we convert it to degrees and add $180°$ . $\newline$ $\theta = \text{atan2}(-6, -7) \times (180/\pi) + 180°$
Perform calculation and round: Perform the calculation and round if necessary. $\newline$ $\theta \approx \text{atan2}(-6, -7) \times (180/\pi) + 180^\circ$ $\newline$ $\theta \approx -139.4^\circ + 180^\circ$ $\newline$ $\theta \approx 40.6^\circ$ $\newline$ Since the angle needs to be between $-180^\circ$ and $180^\circ$ , and our result is $40.6^\circ$ , we do not need to adjust it further.