We are given a complex number $z = 5 - 3i$ and need to find the angle $\theta$ it makes in the complex plane, with $\theta$ expressed in degrees between $-180^\circ$ and $180^\circ$.

Step 1: Recall the formula for the argument (angle) of a complex number

The argument $\theta$ of a complex number $z = a + bi$ can be found using the formula: $\theta = \text{atan2}(b, a)$ where $a$ is the real part and $b$ is the imaginary part.

Step 2: Identify the real and imaginary components

For $z = 5 - 3i$:

• $a = 5$ (real part)
• $b = -3$ (imaginary part)

Step 3: Calculate the angle using the inverse tangent function

We use the formula: $\theta = \text{atan2}(b, a) = \text{atan2}(-3, 5)$ Using a calculator for $\text{atan2}(-3, 5)$, we get:

$\theta \approx -30.96^\circ$

Step 4: Round the result

The problem asks to round to the nearest tenth: $\theta \approx -31.0^\circ$

So, the angle $\theta$ is approximately $-31.0^\circ$.

Would you like more details on this? Here are some related questions you might find interesting:

1. How do you convert an angle in radians to degrees?
2. How do you find the modulus of a complex number?
3. What is the significance of the argument of a complex number in the complex plane?
4. How can you plot complex numbers on the complex plane?
5. What are the uses of polar form for complex numbers?

Tip: The function $\text{atan2}(y, x)$ is more accurate for finding angles than using $\tan^{-1}$ because it takes the sign of both the real and imaginary parts into account, which determines the correct quadrant for the angle.