# Problem based on amplitude of complex number

If the expression $(1 + ir)^3$ is of the form of $s(1 + i)$ for some real $s$ where $r$ is also real and then the value of $r$ can be

$(A) \cot{\frac{\pi}{8}}$ $(B) \tan{\frac{\pi}{12}}$ $(C) \tan{\frac{5\pi}{8}}$ $(D) \sec{\pi}$

Since real and imaginary parts of the given complex number should be same. So I put $3\tan^{-1}(r)=\pi/4$ I'm only getting (B) as an answer but according to my book (B),(C),(D) are all correct options. Where did I go wrong?

• There is no answer (D) ! – Yves Daoust Jul 26 '16 at 8:45

The argument of $(1+ir)^3$ is $3$ times that of $1+ir$, which is known to be one of $\pi/4$, $5\pi/4$. Then $$\arctan(r)=\frac\pi{12},\frac{9\pi}{12},\frac{17\pi}{12},\frac{5\pi}{12},\frac{13\pi}{12},\frac{21\pi}{12}.$$

• Wait wait...I don't get you.The solutions should be $3\tan^{-1}(r)=\pi/4+2\pi n$ right? – user220382 Jul 26 '16 at 8:57
• @ZOZ: $s$ can be negative. – Yves Daoust Jul 26 '16 at 9:02
• Oh silly me.I get it now!Thanks! – user220382 Jul 26 '16 at 9:02

HINT:

$$1-3r^2+i(3r-r^3)=s(1+i)$$

$$\implies3r-r^3=1-3r^2$$

If $r=\tan A\implies\tan3A=\dfrac{3r-r^3}{1-3r^2}=1\implies3A=m\pi+\dfrac\pi4=(4m+1)\dfrac\pi4$ where $m$ is any integer

$A=(4m+1)\dfrac\pi{12}$ where $m\equiv0,1,2\pmod3$

• Nice...but can you tell me the mistake in my method?Why is my method not giving all the solutions? – user220382 Jul 26 '16 at 8:26
• @ZOZ, (Sanchayan) You are taking the principal root only. – lab bhattacharjee Jul 26 '16 at 8:27
• So what should be the correct condition in place of $3\tan^{-1}|r|=\pi/4$ ? – user220382 Jul 26 '16 at 8:28
• Eh!You revealed my pseudonym :-P – user220382 Jul 26 '16 at 8:30
• @ZOZ, Yes, $3\tan^{-1}r=n\pi+\dfrac\pi4$ – lab bhattacharjee Jul 26 '16 at 8:32