Rewrite the following numbers in its polar form r e i θ , where 0 ≤ θ < 2 π . Give exact answers only.

Question

1. 1/5
2. 2+2i
3. -7+7i

Thank you for the help.

Answer 1

1. r = 1/5

θ = 0

Polar Form = 1/5(cos(0)+isin(0)

The polar form of a complex number is z = a+bi and z = r(cosθ +isinθ)

r = |z| = sqrt(a²+b²)

θ = tan^-1(b/a) for a>0

θ = tan^-1(b/a) +180º for a <0



2. r = |z| = sqrt(a²+b²) = sqrt(2²+2²) = sqrt(8)

θ = tan^-1(b/a) = tan^-1(2/2)= 45 rad = π/4

Polar Form = sqrt(8)((cos(π/4)+isin(π/4))

3. r = |z| = sqrt(a²+b²) = sqrt(-7²+7²) = sqrt(98) = 7sqrt(2)

θ = tan^-1(b/a) = tan^-1(-7/7)+180= 135 rad = 3π/4

Polar Form = 7sqrt(2)((cos(3π/4)+isin(3π/4))