In Problems 35-42, find z w and (z)/(w). Leave your answers in polar form. z=4(cos(3 π)/(8)+i si

step 1 Okay, now let's look at this problem. Problem number 39. Okay, so, sorry, problem number 37. So

In Problems 35-42, find z w and (z)/(w). Leave your answers...

In Problems 35-42, find $z w$ and $\frac{z}{w}$. Leave your answers in polar form.
$z=4\left(\cos \frac{3 \pi}{8}+i \sin \frac{3 \pi}{8}\right)$
$w=2\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)$
$w=2\left(\cos \frac{9 \pi}{16}+i \sin \frac{9 \pi}{16}\right)$

Key Concepts

Complex Numbers in Polar Form

Polar form represents complex numbers by expressing them as a product of a modulus (or absolute value) and a trigonometric function of the argument (angle). This form, written as r*(cos(?) + i sin(?)), makes it easier to perform multiplicative and divisive operations and to understand the geometric interpretation of complex numbers in the complex plane.

Multiplication in Polar Form

When multiplying complex numbers in polar form, you multiply their moduli and add their arguments. This property arises from the exponential representation of complex numbers and simplifies computations since it translates the problem into basic arithmetic operations on the moduli and angles separately.

Division in Polar Form

Dividing complex numbers expressed in polar form involves dividing their moduli and subtracting the argument of the divisor from the argument of the dividend. This operation, like multiplication, is simplified by the polar representation, effectively reducing the division of complex numbers to the division of their absolute values and a subtraction of angles.