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Find the argument of the complex number
-1-sqrt3i in the interval
0 <= theta < 2pi. Express your answer in terms of
pi.

Find the argument of the complex number $-1-\sqrt{3} i$ in the interval $0 \leq \theta<2 \pi$ . Express your answer in terms of $\pi$ .

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Find derivatives using logarithmic differentiation

Full solution

Q. Find the argument of the complex number

-1-\sqrt{3} i

in the interval

0 \leq \theta<2 \pi

. Express your answer in terms of

\pi

.

Identify Complex Number: To find the argument of a complex number in the form $a + bi$ , where $a$ is the real part and $b$ is the imaginary part, we need to calculate the angle $\theta$ that the line representing the complex number makes with the positive real axis in the complex plane. The complex number given is $-1 - \sqrt{3}i$ , so $a = -1$ and $b = -\sqrt{3}$ .
Calculate Argument Formula: The argument of a complex number is given by the $\text{arctan}(\frac{b}{a})$ when $a > 0$ . However, when $a < 0$ and $b < 0$ , as in this case, the complex number lies in the third quadrant of the complex plane. The formula for the argument $\theta$ in this case is $\theta = \text{arctan}(\frac{b}{a}) + \pi$ , because we need to account for the additional $\pi$ radians to reach the third quadrant from the positive real axis.
Calculate $\arctan(\frac{b}{a})$ : We calculate the $\arctan(\frac{b}{a})$ using the values of $a$ and $b$ . Here, $\frac{b}{a} = \frac{\sqrt{3}}{1}$ . The $\arctan(\sqrt{3})$ is known to be $\frac{\pi}{3}$ , because $\tan(\frac{\pi}{3}) = \sqrt{3}$ . However, since we are in the third quadrant, we need to add $\pi$ to this value to get the correct argument.
Add pi for Third Quadrant: Adding $\pi$ to $\pi/3$ gives us the argument $\theta$ in the third quadrant. So, $\theta = \pi/3 + \pi = (1/3)\pi + (3/3)\pi = (4/3)\pi$ .
Check Interval: We check to ensure that our argument is within the specified interval $0 \leq \theta < 2\pi$ . Since $(\frac{4}{3})\pi$ is between $0$ and $2\pi$ , our answer is within the correct interval.

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