Exercise 1.5

Theory (before Exercise 1.5)

Polar form (starter)

For a non-zero complex number z = a + bi:

• r = |z| = √(a² + b²)
• θ is an angle such that:

cos θ = a/r
sin θ = b/r

Then:

z = r (cos θ + i sin θ)

Exercise 1.5 (Questions)

Q1 Convert to polar form

Write in the form r(cosθ + i sinθ):

1. z = 3 + 4i
2. z = -1 + √3 i

Q2 Find r and θ (concept)

For z = a + bi with a > 0 and b > 0:

1. What is r?
2. How would you compute θ?

Q3 Mixed check

If z = r(cosθ + i sinθ) show that |z| = r.