Respuesta :
Answer:
The question involves finding the smallest positive value of r such that r[cos(45°) + i sin(45°)] equals z^n for some positive integer n, where z = 3[cos(85°) + i sin(85°)]. Using De Moivre's Theorem, the smallest positive integer n that aligns the angles properly is 8, giving us r = 3^8.
Step-by-step explanation:
The subject of this question is complex numbers in the polar form and deals with finding powers of complex numbers. The student is given z = 3[cos(85°) + i sin(85°)] and is asked to determine the smallest positive value of r such that r [cos(45°) + i sin(45°)] = zn for some positive integer n.
This exploration involves De Moivre's Theorem which states that (cos θ + i sin θ)n = cos(nθ) + i sin(nθ).
By comparing the arguments (angles) and magnitudes (radii) of the two complex numbers in polar forms, the value of n can be found such that the equation holds true.
For r[cos(45°) + i sin(45°)], the value of r should be adjusted to match the magnitude of zn when the angles also coincide. The angle 85° multiplied by n should lead to an angle that is an integer multiple of 360° plus 45°.
The smallest n fulfilling this would be 8, and thus r should be equal to the magnitude of z8, which would be 38.
