Question

If 4 = 5+51 and 22 - 8 Cos3737+ sinthen 222 =28.15 COS197197+ i sin2828OЗлт40,5 cos2819740 cos28+isin197728O2011 (10(195) con (19:)

Answer 1

Given,

The complex number is

`z_1=5+5i`

The another complex number is,

`z_2=8(\cos \text{ }(3\pi)/(7)+i\text{ sin }(3\pi)/(7))`
`\text{The value of z}_1z_2\text{ is,}`

`\begin{gathered} \text{ z}_1\text{z}_2=(5+5i)(8\text{ cos }(3\pi)/(7)+8\text{ i sin }(3\pi)/(7)) \\ \text{z}_1\text{z}_2=(5*8\text{ cos }(3\pi)/(7)+5i*8\text{ cos }(3\pi)/(7)+5*8\text{ i sin }(3\pi)/(7)+5i*8\text{ i sin }(3\pi)/(7)_{}) \\ \end{gathered}`

`\begin{gathered} \text{z}_1\text{z}_2=(40\text{cos }(3\pi)/(7)+40i\text{ cos }(3\pi)/(7)+40\text{ i sin }(3\pi)/(7)-40\text{ sin }(3\pi)/(7)_{}) \\ \text{z}_1\text{z}_2=40(\text{cos }(3\pi)/(7)+i\text{ cos }(3\pi)/(7)+\text{ i sin }(3\pi)/(7)-\text{ sin }(3\pi)/(7)_{}) \end{gathered}`

Coverting cos in to sin and sin in to cos then,

`\begin{gathered} \cos (3\pi)/(7)=\sin \text{ (}(\pi)/(2)-(3\pi)/(7)) \\ \cos (3\pi)/(7)=\sin \text{ (}(\pi)/(14)) \\ \sin (3\pi)/(7)=\cos \text{ (}(\pi)/(14)) \end{gathered}`

Subsituting the values then,

`\begin{gathered} \text{z}_1\text{z}_2=40(\text{cos }(3\pi)/(7)+i\text{ cos }(3\pi)/(7)+\text{ i sin }(3\pi)/(7)-\text{ sin }(3\pi)/(7)_{}) \\ \text{z}_1\text{z}_2=40(\text{cos }(3\pi)/(7)+i\text{ }\sin \text{ }(\pi)/(14)+\text{ i sin }(3\pi)/(7)-\text{ }\cos (\pi)/(14)\text{ }) \\ \end{gathered}`