$z=-6-5 i$ $\newline$ 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=-6-5 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}

Identify Parts of $z$ : Identify the real and imaginary parts of the complex number $z$ . $z = -6 - 5i$ has a real part of $-6$ and an imaginary part of $-5$ .
Calculate Angle using Arctangent: Calculate the angle $\theta$ using the arctangent function. $\newline$ The angle $\theta$ in the complex plane is given by the arctangent of the imaginary part divided by the real part, which is $\text{arctan}(\frac{\text{imaginary}}{\text{real}})$ . $\newline$ $\theta = \text{arctan}(\frac{-5}{-6})$
Use Calculator for Theta: Use a calculator to find the value of theta. $\newline$ $\theta = \text{arctan}(-5/-6) = \text{arctan}(\frac{5}{6})$ $\newline$ Since we are using a calculator, we need to ensure it is set to degree mode.
Calculate Angle in Degrees: Calculate the angle in degrees. $\newline$ $\theta \approx \arctan(\frac{5}{6}) \approx 39.8^\circ$ $\newline$ However, since the complex number is in the third quadrant (both real and imaginary parts are negative), we need to add $180^\circ$ to get the angle in the correct quadrant. $\newline$ $\theta = 39.8^\circ + 180^\circ$
Final Value of Theta: Calculate the final value of theta. $\newline$ $\theta = 39.8° + 180° = 219.8°$ $\newline$ Since we want the angle between $-180°$ and $180°$ , we subtract $360°$ to get the angle in the desired range. $\newline$ $\theta = 219.8° - 360°$
Find Angle in Range: Find the final angle $\theta$ in the specified range. $\theta = -140.2^\circ$ This is the angle that $z$ makes in the complex plane, expressed between $-180^\circ$ and $180^\circ$ .