Write the complex number in trigonometric form $$r ( \cos \theta + i \sin \theta )$$ with θ in the interval [0°, 360°).
-$- 6 - 6 \mathrm { i } \sqrt { 3 }$
A) $12 \left( \cos 290 ^ { \circ } + i \sin 290 ^ { \circ } \right)$
B) $6 \left( \cos 240 ^ { \circ } + i \sin 240 ^ { \circ } \right)$
C) $12 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)$
D) $12 \left( \cos 240 ^ { \circ } + i \sin 240 ^ { \circ } \right)$

Question

Quizplus · Accepted Answer

The magnitude $r$ of the complex number $-6 - 6i\sqrt{3}$ is calculated using the formula $r = \sqrt{a^2 + b^2}$, where $a = -6$ and $b = -6\sqrt{3}$. This gives $r = \sqrt{(-6)^2 + (-6\sqrt{3})^2} = \sqrt{36 + 108} = \sqrt{144} = 12$. The angle $\theta$ is found by taking the arctan of $b/a$, which gives an angle in the third quadrant (since both $a$ and $b$ are negative), specifically at $240^\circ$. Therefore, the correct trigonometric form is $12(\cos 240^\circ + i\sin 240^\circ)$.

Write the Complex Number in Trigonometric Form r(cos⁡θ+isin⁡θ)r ( \cos \theta + i \sin \theta )r(cosθ+isinθ)

Write the Complex Number in Trigonometric Form $r ( \cos \theta + i \sin \theta )$