Find the argument of the complex number 3+sqrt3i in the interval 0 <= theta < 2pi. Express your answer in terms of pi. Answer:

Calculate Angle θ: To find the argument of the complex number 3+sqrt(3)i, we need to calculate the angle θ that the line connecting the origin to the point (3, sqrt(3)) makes with the positive x-axis in the complex plane. The argument is given by θ = arctan(imaginary part / real part).Find Arctan: Calculate the arctan of the imaginary part divided by the real part: arctan(sqrt(3)/3).Identify Triangle Angle: Recognize that arctan(sqrt(3)/3) corresponds to the angle whose tangent is sqrt(3)/3. This is a well-known angle in a 30-60-90 right triangle, where the angle opposite the side with length sqrt(3) is 60 degrees or pi/3 radians.Determine Final Argument: Since the complex number 3+sqrt(3)i is in the first quadrant (both real and imaginary parts are positive), the argument θ is simply pi/3.

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

Find the argument of the complex number $3+\sqrt{3} i$ 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

3+\sqrt{3} i

in the interval

0 \leq \theta<2 \pi

\newline

Express your answer in terms of

\pi

\newline

Answer:

Calculate Angle $\theta$ : To find the argument of the complex number $3+\sqrt{3}i$ , we need to calculate the angle $\theta$ that the line connecting the origin to the point $(3, \sqrt{3})$ makes with the positive x-axis in the complex plane. The argument is given by $\theta = \arctan(\frac{\text{imaginary part}}{\text{real part}})$ .
Find Arctan: Calculate the $\arctan$ of the imaginary part divided by the real part: $\arctan(\sqrt{3}/3)$ .
Identify Triangle Angle: Recognize that $\arctan(\sqrt{3}/3)$ corresponds to the angle whose tangent is $\sqrt{3}/3$ . This is a well-known angle in a $30$ - $60$ - $90$ right triangle, where the angle opposite the side with length $\sqrt{3}$ is $60$ degrees or $\pi/3$ radians.
Determine Final Argument: Since the complex number $3+\sqrt{3}i$ is in the first quadrant (both real and imaginary parts are positive), the argument $\theta$ is simply $\pi/3$ .