Question

Which transformation description best characterizes the transformation of z on the complex plane to obtain w=2(cos(5∘)+isin(5∘))?

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 best description for the transformation of z to obtain w=2(cos(5°)+isin(5°)) is to scale z by a factor of 2 and rotate it 5 degrees counterclockwise. This involves altering both the modulus and argument of the complex number z in the complex plane.

Step-by-step explanation:

The student's question regards the transformation of a complex number z on the complex plane to obtain w=2(cos(5°)+isin(5°)). The transformation involves scaling and rotating z.

By understanding complex numbers in their polar form, we can identify that the complex number w has been obtained by scaling the magnitude of z by a factor of 2 and rotating it by an angle of 5 degrees. The rotation of a complex number by a positive angle in the complex plane is counterclockwise. Therefore, the correct description of the transformation is: Scale z by a factor of 2 and rotate it 5 degrees counterclockwise.

This scaling and rotating refer to the effect on the modulus and argument (angle) of the complex number z respectively, following the formula w=r(cos(θ)+isin(θ)), where r is the modulus and θ is the argument.