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

Find the argument of the complex number $\sqrt{2}-\sqrt{2} i$ 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

\sqrt{2}-\sqrt{2} i

in the interval

0 \leq \theta<2 \pi

. Express your answer in terms of

\pi

.

\newline

Answer:

Calculate Argument using atan $2$ : 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 can use the formula $\theta = \text{atan2}(b, a)$ , where $\text{atan2}$ is the two-argument arctangent function that takes into account the signs of both arguments to determine the correct quadrant of the angle. $\newline$ For the complex number $\sqrt{2} - \sqrt{2}i$ , we have $a = \sqrt{2}$ and $b = -\sqrt{2}$ .
Identify Quadrant: We calculate the argument using the atan $2$ function: $\theta = \text{atan2}(-\sqrt{2}, \sqrt{2})$ . Since both the real and imaginary parts are equal in magnitude but opposite in sign, the argument is in the fourth quadrant.
Calculate Alpha: In the fourth quadrant, the argument of the complex number is $2\pi - \alpha$ , where $\alpha$ is the angle formed with the positive x-axis. Since the real and imaginary parts are equal in magnitude, $\alpha$ is $45$ degrees or $\pi/4$ radians.
Final Argument Calculation: Therefore, the argument $\theta$ is $2\pi - \frac{\pi}{4}$ , which simplifies to $\frac{7\pi}{4}$ .

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