Question

Determine in which quadrant the product of 5(cos3pi/8 + isin 3pi/8) and sqrt 2(cos pi/12 + sin pi/12) lies.

Answer 1

When you multiply two complex numbers given in polar form, the argument of the product is the sum of the arguments of the factors. Meanwhile, the modulus of the product is the product of the moduli of the factors.


In this case, you'd have


`\frac{3\pi}8+\frac\pi{12}=(11\pi)/(24)`

and the modulus would simply be
`5\sqrt2`. Since


`(11\pi)/(24)<(12\pi)/(24)=\frac\pi2`

we would expect the final product to fall in the first quadrant.