8(cos 90° + i sin 90°)/3(cos 30° + i sin 30°)

Question

Find the following quotient, and write the quotient in rectangular form, using exact values.

8(cos 90° + i sin 90°)/3(cos 30° + i sin 30°)

Answer 1

8(cos 90° + i sin 90°)/3(cos 30° + i sin 30°)

=|multiply both numerator and denominator b (cos 30° + i sin 30°),|

  |then (cos 30°+i sin 30°)(cos 30°-i sin 30°)=(cos²30°+sin²30°)=1|

=(8/3)(cos 90° + i sin 90°)(cos 30° - i sin 30°)=

=(8/3)(cos 90°cos 30°+sin 90°sin30°+i(sin 90°cos 90°-sin 30° cos 90°))

=(8/3)(cos(90°-30°)+isin(90°-30°))=(8/3)(cos(60°)+isin(60°))

=(8/3)(1/2+i√3/2)=(4/3)(1+i√3).