Find zw and (z)/(w) . Leave your answers in polar form. z=2(cos(π)/(8)+i sin(π)/(8)) w=2

step 1 Okay, this question wants us to multiply and divide these two complex numbers. So for multiplyin

Find zw and (z)/(w) . Leave your answers in polar form. ...

Find zw and $\frac{z}{w} .$ Leave your answers in polar form.
$$\begin{aligned}
&z=2\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)\\
&w=2\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)
\end{aligned}$$

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Key Concepts

Division of Complex Numbers in Polar Form

For division of complex numbers in polar form, given by r1(cos?1 + i sin?1) divided by r2(cos?2 + i sin?2), the resulting fraction is (r1/r2)(cos(?1 - ?2) + i sin(?1 - ?2)). This method involves dividing the moduli and subtracting the arguments, which greatly simplifies the process compared to division in rectangular form.

Polar Form Representation

Complex numbers can be expressed in polar form as r(cos? + i sin?), where r represents the modulus (or absolute value) and ? represents the argument (or angle) of the complex number. This representation is especially useful in multiplication and division because the operations can be simplified to arithmetic on the moduli and arguments.

Multiplication of Complex Numbers in Polar Form

When multiplying two complex numbers in polar form, such as r1(cos?1 + i sin?1) and r2(cos?2 + i sin?2), the product is given by (r1 * r2)(cos(?1 + ?2) + i sin(?1 + ?2)). This rule utilizes the properties of the trigonometric functions and simplifies the multiplication process by handling moduli and angles separately.

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