Given Z = −1 + 2i, calculate the argument (angle) θ.

Question

ScanSolve AI Tutor · Accepted Answer

✅ Correct.

You have accurately calculated the argument for the complex number $z = -1 + 2i$ .

Here's a breakdown of your steps:

Identify the real and imaginary parts.
For $z = -1 + 2i$ , the real part is $x = -1$ and the imaginary part is $y = 2$ .
Determine the quadrant.
Since $x = -1$ (negative) and $y = 2$ (positive), the complex number lies in the second quadrant.
Calculate the value of $\tan \theta$ .
$\tan \theta = \frac{y}{x} = \frac{2}{-1} = -2$
Find the principal value of $\theta$ from $\arctan(-2)$ .
$\theta = \arctan(-2) \approx -63.43^{\circ}$
This angle is in the fourth quadrant.
Adjust the angle for the correct quadrant.
Since the complex number is in the second quadrant, we need to add $180^{\circ}$ to the angle obtained from $\arctan$ to get the principal argument.
$\theta = -63.43^{\circ} + 180^{\circ}$ $\theta \approx 116.57^{\circ}$
Rounding to one decimal place,
$\theta \approx 116.6^{\circ}$
Your final answer of $\boxed{116.6^{\circ}}$ is correct.

Given Z = -1 + 2i, calculate the argument (angle) θ.