Find the standard form of the complex number shown below. $5 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)$
A) $- \frac { 5 \sqrt { 3 } } { 2 } - \frac { 5 } { 2 }$
B) $\frac { 5 \sqrt { 3 } } { 4 } - \frac { 5 } { 4 }$
C) $\frac { 2 } { 5 } - \frac { 2 \sqrt { 3 } } { 5 }$
D) $\frac { 5 \sqrt { 3 } } { 2 } - \frac { 5 } { 2 }$
E) $\frac { 5 \sqrt { 3 } } { 4 } + \frac { 5 } { 4 }$

Question

Quizplus · Accepted Answer

The standard form of a complex number is $a + bi$, where $a$ and $b$ are real numbers. The given complex number is in polar form, $r(\cos \theta + i\sin \theta)$, where $r = 5$ and $\theta = \frac{11\pi}{6}$. To convert it to standard form, calculate the cosine and sine values: $\cos \frac{11\pi}{6} = \frac{\sqrt{3}}{2}$ and $\sin \frac{11\pi}{6} = -\frac{1}{2}$. Thus, the standard form is $5\left(\frac{\sqrt{3}}{2} - \frac{1}{2}i\right) = \frac{5\sqrt{3}}{2} - \frac{5}{2}i$, which matches option D.

Find the Standard Form of the Complex Number Shown Below 5(cos⁡11π6+isin⁡11π6)5 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)5(cos611π​+isin611π​)

Find the Standard Form of the Complex Number Shown Below $5 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)$