Final answer:
The expression z=(cos θ+isinθ)⁵/(cos θ-isinθ)⁴ simplifies to ei9θ with a modulus of 1 and an amplitude of 9θ.
Step-by-step explanation:
The student asked to simplify the expression z=(cos θ+isinθ)⁵/(cos θ-isinθ)⁴ into x+iy form and to find its modulus and amplitude.
We can use Euler's formula, eiδ = cos(β) + i sin (β), to express cos θ + i sin θ as eiθ. Thus the given expression can be rewritten as z=(eiθ)⁵/(e-iθ)⁴ which simplifies to z=eiθ⁹. The modulus of z is 1 since it's a complex number on the unit circle. The amplitude is θ multiplied by 9 because the argument of z is 9θ, which reflects 9 times the angle θ around the origin.