Question

angle of complex numbers
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you might need: calculator
$z = 5 - 3i$
find the angle $\theta$ (in degrees) that $z$ makes in the complex plane.
round your answer, if necessary, to the nearest tenth. express $\theta$ between $-180^\circ$ and $180^\circ$.
$\theta = $ °

Explanation:

Step1: Determine the quadrant

For the complex number \(z = 5-3i\), the real part \(a = 5>0\) and the imaginary part \(b=-3 < 0\). So, \(z\) lies in the fourth quadrant.

Step2: Use the formula for the argument

The formula for the argument \(\theta\) of a complex number \(z=a + bi\) is \(\theta=\arctan(\frac{b}{a})\) (adjusted for the quadrant). Here, \(a = 5\) and \(b=-3\). So, \(\theta=\arctan(\frac{-3}{5})\).
Using a calculator, \(\arctan(\frac{3}{5})\approx30.96^{\circ}\). Since \(z\) is in the fourth quadrant, \(\theta=- 30.96^{\circ}\approx - 31.0^{\circ}\)

Answer:

\(-31.0\)