Question Number 185821 by Rupesh123 last updated on 28/Jan/23
Answered by qaz last updated on 28/Jan/23
![5arctan (1/7)+2arctan (3/(79))=5Arg(7+i)+2Arg(79+3i) =Arg[(7+i)^5 ∙(79+3i)^2 ]=Arg[78125000(1+i)] =arctan ((78125000)/(78125000))=(π/4)](https://www.tinkutara.com/question/Q185822.png)
$$\mathrm{5arctan}\:\frac{\mathrm{1}}{\mathrm{7}}+\mathrm{2arctan}\:\frac{\mathrm{3}}{\mathrm{79}}=\mathrm{5}{Arg}\left(\mathrm{7}+{i}\right)+\mathrm{2}{Arg}\left(\mathrm{79}+\mathrm{3}{i}\right) \\ $$$$={Arg}\left[\left(\mathrm{7}+{i}\right)^{\mathrm{5}} \centerdot\left(\mathrm{79}+\mathrm{3}{i}\right)^{\mathrm{2}} \right]={Arg}\left[\mathrm{78125000}\left(\mathrm{1}+{i}\right)\right] \\ $$$$=\mathrm{arctan}\:\frac{\mathrm{78125000}}{\mathrm{78125000}}=\frac{\pi}{\mathrm{4}} \\ $$