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Find the argument of the complex number
2+2i in the interval
0 <= theta < 2pi. Express your answer in terms of
pi.

Find the argument of the complex number $2+2 i$ in the interval $0 \leq \theta<2 \pi$ . Express your answer in terms of $\pi$ .

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Find derivatives of trigonometric functions using the product and quotient rules

Full solution

Q. Find the argument of the complex number

2+2 i

in the interval

0 \leq \theta<2 \pi

. Express your answer in terms of

\pi

.

Understand Complex Number Argument: Understand the concept of the argument of a complex number. The argument of a complex number $z = a + bi$ , where $a$ and $b$ are real numbers, is the angle $\theta$ in polar coordinates that the line connecting the origin to the point $(a, b)$ makes with the positive real axis. The argument is usually denoted as $\text{arg}(z)$ and is measured in radians.
Convert to Polar Form: Convert the complex number to polar form to find the argument. $\newline$ For the complex number $2+2i$ , we can find the argument by calculating the angle $\theta$ using the formula $\theta = \arctan(\frac{b}{a})$ , where $a$ is the real part and $b$ is the imaginary part of the complex number.
Calculate Using Arctan: Calculate the argument using the arctan function. $\newline$ For the complex number $2+2i$ , $a = 2$ and $b = 2$ . Therefore, $\theta = \text{arctan}(\frac{2}{2}) = \text{arctan}(1)$ .
Evaluate Arctan of $1$ : Evaluate the arctan of $1$ . $\newline$ The arctan of $1$ is $\frac{\pi}{4}$ radians because the tangent of $\frac{\pi}{4}$ is $1$ . This is a well-known value from trigonometry.
Ensure Correct Interval: Ensure the argument is in the correct interval. $\newline$ Since we are looking for the argument in the interval $0 \leq \theta < 2\pi$ , and $\pi/4$ is already in this interval, we do not need to adjust the value of the argument.
Write Final Answer: Write the final answer. $\newline$ The argument of the complex number $2+2i$ is $\pi/4$ radians.

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