How do you simplify 1/2(cos(pi/3)+isin(pi/3))*6(cos((5pi)/6)+isin((5pi)/6)) and express the result in rectangular form?

May 11, 2017

To multiply, multiply the magnitudes and add the angles.
Substitute the equivalents for the trigonometric functions.
Use the distributive property.

Explanation:

To multiply, multiply the magnitudes and add the angles.

$\frac{1}{2} \left(\cos \left(\frac{\pi}{3}\right) + i \sin \left(\frac{\pi}{3}\right)\right) \cdot 6 \left(\cos \left(\frac{5 \pi}{6}\right) + i \sin \left(\frac{5 \pi}{6}\right)\right) = 3 \left(\cos \left(\frac{7 \pi}{6}\right) + i \sin \left(\frac{7 \pi}{6}\right)\right)$

Substitute the equivalents for the trigonometric functions.

$\frac{1}{2} \left(\cos \left(\frac{\pi}{3}\right) + i \sin \left(\frac{\pi}{3}\right)\right) \cdot 6 \left(\cos \left(\frac{5 \pi}{6}\right) + i \sin \left(\frac{5 \pi}{6}\right)\right) = 3 \left(\frac{\sqrt{3}}{2} - \frac{1}{2} i\right)$

Use the distributive property:

$\frac{1}{2} \left(\cos \left(\frac{\pi}{3}\right) + i \sin \left(\frac{\pi}{3}\right)\right) \cdot 6 \left(\cos \left(\frac{5 \pi}{6}\right) + i \sin \left(\frac{5 \pi}{6}\right)\right) = \frac{3 \sqrt{3}}{2} - \frac{3}{2} i$