Question

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?

Answer 1

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.

`1/2(cos(pi/3)+isin(pi/3))*6(cos((5pi)/6)+isin((5pi)/6)) = 3(cos((7pi)/6)+isin((7pi)/6))`

Substitute the equivalents for the trigonometric functions.

`1/2(cos(pi/3)+isin(pi/3))*6(cos((5pi)/6)+isin((5pi)/6)) = 3(sqrt3/2-1/2i)`

Use the distributive property:

`1/2(cos(pi/3)+isin(pi/3))*6(cos((5pi)/6)+isin((5pi)/6)) = (3sqrt3)/2-3/2i`