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{ }\frac{3\pi}{7}+i\text{ sin }\frac{3\pi}{7})$$\text{The value of z}_1z_2\text{ is,}$

$\begin{gathered} \text{ z}_1\text{z}_2=(5+5i)(8\text{ cos }\frac{3\pi}{7}+8\text{ i sin }\frac{3\pi}{7}) \\ \text{z}_1\text{z}_2=(5\times8\text{ cos }\frac{3\pi}{7}+5i\times8\text{ cos }\frac{3\pi}{7}+5\times8\text{ i sin }\frac{3\pi}{7}+5i\times8\text{ i sin }\frac{3\pi}{7}_{}) \\ \end{gathered}$

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

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

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

Subsituting the values then,

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