Find trigonometric and algebric

Question

Let Z1= 10(cos(theta) + i sin(theta)) , Z2= 1/2 (sin(2theta) + i cos(2theta)), where tan(theta)= 3/4, theta belongs to [0°, 90° [ Find trigonometric and algebric forms of the product Z1Z2.

Answer 1

Represent theta by t:

tan(t)=3/4 so sin(t)=3/5, cos(t)=4/5.

Z1Z2=5(cos(t)sin(2t)-sin(t)cos(2t)+i(cos(t)cos(2t)+sin(t)sin(2t))=

5(sin(2t-t)+icos(2t-t))=5(sin(t)+icos(t))=5(3+4i)/5=3+4i.

This complex result can be represented by a right-angled triangle with sides 3, 4, 5 (hypotenuse).

Theta is the smaller of the two other angles of the triangle.