The complex Plane and Polar form for Complex Numbers?

Question

Perform the indicated multiplication or division; express your answer in both rectangular form a+bi and polar form r(cosΘ + i sin Θ).

1.

√54(cos 9pie/4 + i sin pie/10)
---------------------------------------------------
√6(cos 7pie/12 + i sin 7pie/12)

Please show all formulas and work.

Thank you.

Answer 1

π=pi, not "pie". "Pie" is a tasty dessert, especially pumpkin pie. Moving on:

1: In order to divide numbers in polar form, you divide the magnitudes and subtract the angles. I'm assuming that "i sin pie/10" was a typo, and you meant to put i sin (9π/4), which would match the angle for cosine. Assuming that is the case, the magnitude would be √54/√6 = √9 = 3, and the angle would be 9π/4 - 7π/12 = 27π/12 - 7π/12 = 20π/12 = 5π/3. So the quotient will be:

3 (cos (5π/3) + i sin (5π/3))

Or in polar form, since cos (5π/3) = cos (-π/3) = 1/2 and sin (5π/3) = -√3/2, we have:

3 (1/2 - i√3/2)
3/2 - 3i√3/2

And we are done.

Answer 2

i) 1 - i √3, r^2=(1+3), so r=2 2(1/2-i(√3)/2)=2(cos Θ + i sin Θ) where Θ=-pi/3 (ii) - i + i √3 Same method as above 2(cos Θ + i sin Θ) where Θ=2pi/3 (iii) - 1 - i=√2(-1/√2 -i/√2)=√2((cos Θ + i sin Θ) where Θ=-3pi/4