Find the argument of the complex number 6-2sqrt3i in the interval 0 <= theta < 2pi. Express your answer in terms of pi. Answer:

Calculate arctan value: 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 use the formula theta = arctan(b/a). The complex number given is 6 - 2sqrt3i, so a = 6 and b = -2sqrt3.Simplify arctan expression: First, we calculate the arctan of b/a, which is arctan(-2sqrt3/6). Simplifying the fraction gives us arctan(-sqrt3/3).Adjust angle for interval: The value of arctan(-sqrt3/3) corresponds to the angle -pi/6, because tan(-pi/6) = -sqrt3/3. However, since we want the argument in the interval 0 <= theta < 2pi, we need to add 2pi to the negative angle to find the equivalent positive angle.Finalize argument value: Adding 2pi to -pi/6 gives us the angle 11pi/6, which is in the desired interval 0 <= theta < 2pi.

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Find the argument of the complex number
6-2sqrt3i in the interval
0 <= theta < 2pi.
Express your answer in terms of
pi.
Answer:

Find the argument of the complex number $6-2 \sqrt{3} i$ in the interval 0 \leq \theta<2 \pi . $\newline$ Express your answer in terms of $\pi$ . $\newline$ Answer:

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Q. Find the argument of the complex number

6-2 \sqrt{3} i

in the interval

0 \leq \theta<2 \pi

\newline

Express your answer in terms of

\pi

\newline

Answer:

Calculate arctan value: 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 use the formula $\theta = \text{arctan}(\frac{b}{a})$ . The complex number given is $6 - 2\sqrt{3}i$ , so $a = 6$ and $b = -2\sqrt{3}$ .
Simplify arctan expression: First, we calculate the arctan of $b/a$ , which is $\text{arctan}(-2\sqrt{3}/6)$ . Simplifying the fraction gives us $\text{arctan}(-\sqrt{3}/3)$ .
Adjust angle for interval: The value of $\arctan(-\sqrt{3}/3)$ corresponds to the angle $-\pi/6$ , because $\tan(-\pi/6) = -\sqrt{3}/3$ . However, since we want the argument in the interval 0 \leq \theta < 2\pi, we need to add $2\pi$ to the negative angle to find the equivalent positive angle.
Finalize argument value: Adding $2\pi$ to $-\pi/6$ gives us the angle $11\pi/6$ , which is in the desired interval 0 \leq \theta < 2\pi.