Express z_(1)=-10sqrt3-10 i 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)=-10sqrt3-10 i 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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Calculate Magnitude: To express the complex number $ z_1 = -10\sqrt{3} - 10i $ in polar form, we need to find its magnitude (r) and angle ($\theta$) with respect to the positive x-axis. The magnitude is found using the formula $ r = \sqrt{a^2 + b^2} $, where $ a $ is the real part and $ b $ is the imaginary part of the complex number. Calculation: $ r = \sqrt{(-10\sqrt{3})^2 + (-10)^2} = \sqrt{300 + 100} = \sqrt{400} = 20 $.Calculate Angle: Next, we find the angle $\theta$ using the formula $ \theta = \arctan\left(\frac{b}{a}\right) $, where $ a $ is the real part and $ b $ is the imaginary part of the complex number. Since the complex number is in the third quadrant (both real and imaginary parts are negative), we need to add $\pi$ to the angle obtained from the arctan function to get the angle in the correct quadrant. Calculation: $ \theta = \arctan\left(\frac{-10}{-10\sqrt{3}}\right) = \arctan\left(\frac{1}{\sqrt{3}}\right) $. The arctan of $ \frac{1}{\sqrt{3}} $ is $ \frac{\pi}{6} $, but since the complex number is in the third quadrant, we add $\pi$ to get $ \theta = \frac{\pi}{6} + \pi = \frac{7\pi}{6} $.Find Polar Form: The polar form of a complex number is given by $ r(\cos(\theta) + i\sin(\theta)) $. Substituting the values of $ r $ and $ \theta $ we found in the previous steps, we get the polar form of $ z_1 $. Calculation: $ z_1 = 20(\cos(\frac{7\pi}{6}) + i\sin(\frac{7\pi}{6})) $.

Express $z_{1}=-10 \sqrt{3}-10 i$ 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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Express $z_{1}=-10 \sqrt{3}-10 i$ in polar form. $\newline$ Express your answer in exact terms, using radians, where your angle is between $0$ and $2 \pi$ radians, inclusive. $\newline$ $z_{1}=$