Unit Test
Unit Test
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For the complex number $z=\frac{5\sqrt{3}}{4}-\frac{5}{4}i$, what is the polar form?
$z=\frac{5}{2}(\cos \left(\frac{\pi }{6}\right)+i\sin \left(\frac{\pi }{6}\right)$
$z=\frac{5}{4}(\cos \left(\frac{\pi }{6}\right)+i\sin \left(\frac{\pi }{6}\right)$
$z=\frac{5}{2}(\cos \left(\frac{11\pi }{6}\right)+i\sin \left(\frac{11\pi }{6}\right)$
$z=\frac{5}{4}(\cos \left(\frac{11\pi }{6}\right)+i\sin \left(\frac{11\pi }{6}\right)$
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Step 1 ：The question is asking for the polar form of a complex number. The polar form of a complex number is given by $r(\cos (\theta )+i\sin (\theta ))$ where $r$ is the magnitude of the complex number and $\theta$ is the angle it makes with the real axis.

Step 2 ：The magnitude of a complex number $a+bi$ is given by $\sqrt{a^2+b^2}$ and the angle is given by $\arctan (\frac{b}{a})$ if $a>0$ and $\arctan (\frac{b}{a})+\pi$ if $a<0$.

Step 3 ：In this case, the complex number is $\frac{5\sqrt{3}}{4}-\frac{5}{4}i$. So, $a=\frac{5\sqrt{3}}{4}$ and $b=-\frac{5}{4}$.

Step 4 ：Let's calculate the magnitude and the angle.

Step 5 ：The magnitude of the complex number is 2.5 and the angle is -0.5235987755982989 radians. However, the angle is negative which means the complex number is in the fourth quadrant. To convert this to a positive angle, we add $2\pi$ to the angle.

Step 6 ：Also, the options for the question are in terms of $\pi$. So, we need to express the angle in terms of $\pi$.

Step 7 ：The magnitude of the complex number is 2.5 and the angle is $\frac{11\pi }{6}$.

Step 8 ：Looking at the options, the correct polar form of the complex number is $z=\frac{5}{2}(\cos \left(\frac{11\pi }{6}\right)+i\sin \left(\frac{11\pi }{6}\right)$

Step 9 ：$\boxed{{\displaystyle z=\frac{5}{2}(\cos \left(\frac{11\pi }{6}\right)+i\sin \left(\frac{11\pi }{6}\right)}}$