In Problems 35-42, find z w and (z)/(w). Leave your answers in polar form. z=3(cos130^∘+i s

Step 1: Express the complex numbers u005C( z u005C) and u005C( w u005C) in polar form.u000Au002D u005C( z u003D 3(u005Ccos 130^u005Ccirc

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.
$$
\begin{aligned}
& z=3\left(\cos 130^{\circ}+i \sin 130^{\circ}\right) \\
& w=4\left(\cos 270^{\circ}+i \sin 270^{\circ}\right)
\end{aligned}
$$

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Key Concepts

Multiplication of Complex Numbers in Polar Form

When multiplying two complex numbers in polar form, you multiply their magnitudes and add their angles. If z = r1(cos ?1 + i sin ?1) and w = r2(cos ?2 + i sin ?2), then their product zw is given by (r1 * r2)(cos(?1 + ?2) + i sin(?1 + ?2)). This property significantly simplifies the multiplication process compared to the rectangular form.

Division of Complex Numbers in Polar Form

Dividing complex numbers in polar form involves dividing their magnitudes and subtracting the angles. For two complex numbers, z and w, with z = r1(cos ?1 + i sin ?1) and w = r2(cos ?2 + i sin ?2), the quotient z/w is calculated as (r1/r2)(cos(?1 - ?2) + i sin(?1 - ?2)). This method makes the division of complex numbers straightforward by operating separately on the magnitudes and angles.

Complex Numbers in Polar Form

Complex numbers can be expressed in polar form as r(cos ? + i sin ?), where r is the magnitude (modulus) and ? is the argument (angle) of the complex number. This form is particularly useful for multiplication, division, and exponential operations as it simplifies computations involving angles and magnitudes.

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