Question

find the following write your answers in trigonometry form give exact values in your answers not decimal approximations

Answer 1

Given:

z1 = 6(cos80 + isin80)

z2 = 8(cos140 + isin140)

Let's solve for the following:

(a) z1/z2

To solve for z1/z2, we have:

`\begin{gathered} (z_1)/(z_2)=(6(\cos 80+i\sin 80))/(8(\cos 140+i\sin 140)) \\ \\ (z_1)/(z_2)=(3(\cos 80+i\sin 80))/(4(\cos 140+i\sin 140)) \end{gathered}`

Solving further:

`(z_1)/(z_2)=(0.521+2.954i)/(-3.064+2.571i)`

Multiply the denominator and numerator by the conjugate:

`\begin{gathered} (0.521+2.954i)/(-3.064+2.571i)*(-3.064-2.571i)/(-3.064-2.571i) \\ \\ ((0.521+2.954i)(-3.064-2.571i))/((-3.064+2.571i)(-3.064-2.571i)) \\ \\ =(6-10.392i)/(16) \\ \\ =(1)/(16)\ast(6-10.392i)/(1) \end{gathered}`

Let's write in trigonometric form:

`\begin{gathered} 0.0625(6-10.392i) \\ \\ 0.0625(6)+0.0625(-10.392i) \\ \\ 0.375-0.649i \\ \\ \lvert z\rvert=\sqrt[]{(-0.649)^2+(}0.375)^2 \\ \\ \lvert z\rvert=\sqrt[]{0.562}=0.749 \\ \\ \tan ^(-1)((-0.649)/(0.375))=-60^o \end{gathered}`

Therefore, the answer in trigonometric form is:

`0.749(\cos (-60^0)+i\sin (-60))`

Part b.

`z_1z_2=(6(\text{cos}80+i\sin 80))*(8(\cos 140+i\sin 140))`

Thus, we have:

`\begin{gathered} z_1z_2=\mleft(1.042+5.909i\mright)*(-6.128+5.142i) \\ \\ \end{gathered}`

Apply the FOIL method:

`1.042(-6.128+5.142i)+5.909i(-6.128+5.142i)`

Apply distributive property: