$z=5-3 i$ $\newline$ 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=5-3 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 $z = 5 - 3i$ makes with the positive x-axis in the complex plane. This angle is also known as the argument of the complex number.
Convert to polar form: Convert the complex number to polar form. $\newline$ To find the angle $\theta$ , we can convert the complex number from rectangular form $(a + bi)$ to polar form $(r(\cos(\theta) + i\sin(\theta)))$ . The angle $\theta$ is the $\arctan$ of the imaginary part divided by the real part.
Calculate angle with arctan: Calculate the angle using the arctan function. $\newline$ $\theta = \text{arctan}(-\frac{3}{5})$ $\newline$ We use the arctan function because it gives us the angle whose tangent is the quotient of the imaginary part and the real part of the complex number.
Use calculator for angle: Use a calculator to find the angle. $\theta = \arctan(-\frac{3}{5}) \approx \arctan(-0.6)$ Using a calculator, we find that $\theta \approx -30.96$ degrees.
Adjust angle within range: Adjust the angle to be within the specified range. $\newline$ Since the angle must be between $-180^\circ$ and $180^\circ$ , and our calculated angle is already within this range, no further adjustment is necessary.
Round angle if necessary: Round the angle to the nearest tenth, if necessary. $\theta \approx -30.96$ degrees $\approx -31.0$ degrees when rounded to the nearest tenth.