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Find the argument of the complex number
3sqrt2+3sqrt2i in the interval
0 <= theta < 2pi. Express your answer in terms of
pi.
Answer:

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

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Find trigonometric ratios using a Pythagorean or reciprocal identity

Full solution

Q. Find the argument of the complex number

3 \sqrt{2}+3 \sqrt{2} i

in the interval

0 \leq \theta<2 \pi

. Express your answer in terms of

\pi

.

\newline

Answer:

Given Complex Number: We are given a complex number in the form $a + bi$ , where $a = 3\sqrt{2}$ and $b = 3\sqrt{2}$ . The argument of a complex number is the angle $\theta$ in polar coordinates that the vector representing the complex number makes with the positive real axis. The argument can be found using the $\text{arctan}$ function, which gives the angle in radians.
Determine Quadrant: Since both the real and imaginary parts of the complex number are positive and equal, the complex number lies on the line $y = x$ in the first quadrant of the complex plane. This means that the argument $\theta$ is $45$ degrees or $\frac{\pi}{4}$ radians. However, we need to ensure that this is the correct angle for the given interval $0 \leq \theta < 2\pi$ .
Convert Degrees to Radians: To convert $45$ degrees to radians, we use the fact that $\pi$ radians is equal to $180$ degrees. Therefore, $45$ degrees is $(45/180) \times \pi = \pi/4$ radians. Since this angle is in the first quadrant and within the specified interval, it is the correct argument for the complex number.
Confirm Correctness: We can confirm that there is no math error by checking that the real and imaginary parts are equal and positive, which places the complex number in the first quadrant, and that the angle we found is indeed in the interval $0 \leq \theta < 2\pi$ .

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