Find the argument of the complex number -1-sqrt3i in the interval 0

Question

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

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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 arctan(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 = arctan(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(b/a): We calculate the arctan(b/a) using the values of a and b. Here, b/a = sqrt(3)/1. The arctan(sqrt(3)) is known to be pi/3, because tan(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 <= theta < 2pi. Since (4/3)pi is between 0 and 2pi, our answer is within the correct interval.

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

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 $\pi$ .