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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.

Find the angle θ (in degrees) that z = 5 - 3i makes in the complex plane. Round your answer, if necessary, to the nearest tenth. Express θ between -180° and 180°.

Learn how to find the argument (angle) of the complex number z = 5 - 3i in the complex plane. This problem explores key concepts in complex numbers, such as the use of the atan2 function for calculating angles, and is suitable for students in Grades 10-12.

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