Question

find the quotient, and write it in rectangular form. \\(\frac{8(\cos 300^{\circ} + i\sin 300^{\circ})}{2(\cos 120^{\circ} + i\sin 120^{\circ})} = \square\\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. type your answer in the form \\(a + bi\\).)

Explanation:

Step1: Use De Moivre's Theorem for Division

For complex numbers in polar form \( r_1(\cos\theta_1 + i\sin\theta_1) \) and \( r_2(\cos\theta_2 + i\sin\theta_2) \), the quotient is \( \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)] \). Here, \( r_1 = 8 \), \( \theta_1 = 300^\circ \), \( r_2 = 2 \), \( \theta_2 = 120^\circ \). So first, calculate the modulus quotient: \( \frac{8}{2} = 4 \). Then the angle difference: \( 300^\circ - 120^\circ = 180^\circ \).

Step2: Simplify the Trigonometric Values

We know that \( \cos 180^\circ = -1 \) and \( \sin 180^\circ = 0 \). Substitute these into the polar form of the quotient: \( 4(\cos 180^\circ + i\sin 180^\circ) = 4(-1 + i\cdot 0) \).

Step3: Convert to Rectangular Form

Simplify the expression: \( 4(-1) + 4(0)i = -4 + 0i \).

Answer:

\( -4 + 0i \) (or simply \( -4 \) since the imaginary part is 0)