write $z = 2\\left(\\cos \\frac{5\\pi}{6} + i\\sin \\frac{5\\pi}{6}\\right)$ in rectangular form.\n(1…

write $z = 2\\left(\\cos \\frac{5\\pi}{6} + i\\sin \\frac{5\\pi}{6}\\right)$ in rectangular form.\n(1 point)\n$\\bigcirc$ $z = 1 - i\\sqrt{3}$\n$\\bigcirc$ $z = -\\frac{\\sqrt{3}}{2} + i\\frac{1}{2}$\n$\\bigcirc$ $z = -i\\sqrt{3} + 1$\n$\\bigcirc$ $z = -\\sqrt{3} + i$

write $z = 2\\left(\\cos \\frac{5\\pi}{6} + i\\sin \\frac{5\\pi}{6}\\right)$ in rectangular form.\n(1 point)\n$\\bigcirc$ $z = 1 - i\\sqrt{3}$\n$\\bigcirc$ $z = -\\frac{\\sqrt{3}}{2} + i\\frac{1}{2}$\n$\\bigcirc$ $z = -i\\sqrt{3} + 1$\n$\\bigcirc$ $z = -\\sqrt{3} + i$

Answer

Explanation:

Step1: Evaluate $\cos\frac{5\pi}{6}$

$\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}$

Step2: Evaluate $\sin\frac{5\pi}{6}$

$\sin\frac{5\pi}{6}=\frac{1}{2}$

Step3: Distribute the coefficient 2

$z=2\times\left(-\frac{\sqrt{3}}{2}\right) + i\times\left(2\times\frac{1}{2}\right)$

Step4: Simplify each term

$z=-\sqrt{3} + i$

Answer:

$z=-\sqrt{3}+i$