Question

Which description of the transformation of z on the complex plane gives the quotient of and w = 2 (cosine (5 degrees) i sine (5 degrees) )?

a. Scale z by a factor of 2 and rotate it 5 degrees clockwise
b. Scale z by a factor of one-half and rotate it 5 degrees clockwise
c. Scale z by a factor of 2 and rotate it 5 degrees counterclockwise
d. Scale z by a factor of one-half and rotate it 5 degrees counterclockwise

Answer 1

The quotient of a complex number z by w = 2 (cos(5) + i sin(5)) scales z by one-half and rotates it 5 degrees counterclockwise.

Step-by-step explanation:

The quotient of a complex number z by w = 2 (cosine (5 degrees) + i sine (5 degrees)) results in a transformation of z on the complex plane. As w has a modulus of 2 and an argument of 5 degrees, dividing by w will scale z by the reciprocal of 2, which is one-half, and rotate z by the negative of w's argument. Therefore, the transformation will be a scaling factor of one-half and a rotation of 5 degrees counterclockwise.