Precalculus Questions

Question

1 .Write the complex number in polar form with argument θ between 0 and 2π.

1 + 3 i

5i(1 + i)

2.Find the product z₁z₂ and the quotient z_1/z_2. Express your answers in polar form.

z1 = 2(cos(5π/4) + i sin(5π/4),  z2 = 5(cos(π) + i sin(π))

z₁z₂₌

z_1/z_2.=

3. Find the indicated power using De Moivre's Theorem. (Express your fully simplified answer in the form a + bi.).

(2/2 + 2/2 i)8

(1 − i)18

Answer 1

1+√3

r=√(1^2+√3^2)=√(1+3)=2

tanθ=√3/1=√3, so θ=60°

Z=2(cos60°+isin60°)

5i(1+i)=5i+5i^2=-5+5i

r=√50=5√2

tanθ=5/(-5)=-1, so θ=135°

Z=5√2(cos135°+isin135°)

now multiply the radii and add the angles

z1z2=5√2(cos[9∏/4) +isin[9∏/4])

now divide the radii and subtract the angles

z1/z2=(√2/5)(cos[∏/4]+isin[∏/4])

change to polar coordinates as above

r=√((√2/2)^2+(√2/2)^2)=√((2/4)+(2/4))=√1=1

tanθ=1, so θ=45°

Z^8=1^8(cos[8*45]+isin[8*45])

Z^8=1(cos360°+isin360°)

Z^8=1(1+i*0)

Z^8=1

try the last one

you should get...

Z^18=[√2]^18(cos[18*135]+isin18*135])

18*135=2430, now see how many 360s there are in 2430 and subtract them to get the degree measure