Precalculus Trigonometry Divide and express the result in s

Precalculus & Trigonometry

Divide and express the result in standard notation a + bi: 2(cos pi/3 + I sin pi/3)/6(cos pi/6 + I sin pi/6).

Solution

2(cos pi/3 + i sin pi/3)/6(cos pi/6 + isinpi/6)

First we have to convert them in exponential form.

r(cos theta + i sin theta)= r e^{i theta} where theta is in radian

therefore 2(cos pi/3 + i sin pi/3)= 2 e^{i pi/3}

6(cos pi/6 + isinpi/6)= 6 e^ipi/6

so we get

2e^{i pi/3}/6e^{i pi/6} = (1/3 ) e^{i(pi/3 -pi/6)}= (1/3)e^{i pi/6}

(1/3)(cos pi/6 + i sin pi/6) = 1/3(sqrt3/2 + i/2)= sqrt3/6 + 1i/6