Express z_(1)=3sqrt3-9i in polar form. Express your answer in exact terms, using radians, where your angle is between 0 and 2pi radians, inclusive. z_(1)=

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Express  z_(1)=3sqrt3-9i in polar form. Express your answer in exact terms, using radians, where your angle is between 0 and  2pi radians, inclusive.  z_(1)=

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Magnitude Calculation: To express the complex number $ z_1 = 3\sqrt{3} - 9i $ in polar form, we need to find its magnitude (r) and angle ($\theta$) with respect to the positive x-axis. The polar form is given by $ r(\cos(\theta) + i\sin(\theta)) $ or $ re^{i\theta} $.Angle Calculation: First, calculate the magnitude $ r $ using the formula $ r = \sqrt{a^2 + b^2} $, where $ a = 3\sqrt{3} $ is the real part and $ b = -9 $ is the imaginary part of the complex number. $ r = \sqrt{(3\sqrt{3})^2 + (-9)^2} = \sqrt{27 + 81} = \sqrt{108} = 6\sqrt{3} $.Angle Calculation Continued: Next, we need to find the angle $ \theta $. The angle is determined by the arctangent of the imaginary part divided by the real part, $ \theta = \arctan\left(\frac{b}{a}\right) $. However, since the complex number is in the fourth quadrant (real part positive, imaginary part negative), we need to add $ 2\pi $ to the angle to ensure it is between 0 and $ 2\pi $. $ \theta = \arctan\left(\frac{-9}{3\sqrt{3}}\right) + 2\pi $.Polar Form Calculation: Calculate the angle $ \theta $ using the values of $ a $ and $ b $. $ \theta = \arctan\left(\frac{-9}{3\sqrt{3}}\right) + 2\pi = \arctan\left(\frac{-9}{3\sqrt{3}}\right) + 2\pi = \arctan\left(\frac{-3}{\sqrt{3}}\right) + 2\pi = \arctan(-\sqrt{3}) + 2\pi $. Since $ \arctan(-\sqrt{3}) $ corresponds to $ -\frac{\pi}{3} $, we have: $ \theta = -\frac{\pi}{3} + 2\pi = \frac{5\pi}{3} $.Now we can write the complex number in polar form using the magnitude $ r $ and angle $ \theta $ we found. $ z_1 = 6\sqrt{3}(\cos(\frac{5\pi}{3}) + i\sin(\frac{5\pi}{3})) $.

Express $z_{1}=3 \sqrt{3}-9 i$ in polar form. $\newline$ Express your answer in exact terms, using radians, where your angle is between $0$ and $2 \pi$ radians, inclusive. $\newline$ $z_{1}=$

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