$z=-3-8 i$ $\newline$ 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 $180^{\circ}$ . $\newline$ $\theta=\square^{\circ}$

Full solution

z=-3-8 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 real and imaginary parts: Identify the real and imaginary parts of the complex number $z$ . $z = -3 - 8i$ , where the real part is $-3$ and the imaginary part is $-8$ .
Calculate the argument of z: Calculate the argument of z, which is the angle $\theta$ in the complex plane. $\newline$ The argument of z is given by $\theta = \arctan(\frac{\text{imaginary part}}{\text{real part}}) = \arctan(\frac{-8}{-3})$ .
Use calculator to find theta in radians: Use a calculator to find the value of $\theta$ in radians. $\theta = \text{arctan}(-8 / -3) = \text{arctan}(8 / 3)$ .Since we need the angle in degrees, we will convert it after calculation.
Convert radians to degrees: Convert the angle from radians to degrees. $\theta$ in degrees = $\arctan(\frac{8}{3}) \times (\frac{180}{\pi})$ .
Calculate theta in degrees: Calculate the angle $\theta$ in degrees using a calculator. $\theta$ in degrees $\approx \arctan(\frac{8}{3}) \times (\frac{180}{\pi}) \approx 69.4^\circ$ . However, since both the real and imaginary parts of $z$ are negative, $z$ lies in the third quadrant, and the angle should be adjusted to be between $-180^\circ$ and $180^\circ$ .
Adjust angle for correct quadrant: Adjust the angle for the correct quadrant. $\newline$ In the third quadrant, the angle $\theta$ should be negative and we add $180°$ to the calculated angle to get the angle in the correct range. $\newline$ $\theta = 69.4° - 180° = -110.6°$ .