$i$ won't stop

Algebra Level pending

True or false:

1) $\sqrt{i} = cos(\frac{\pi}{4}) + sin(\frac{\pi}{4})$

2) $\sqrt{i} = cos(\frac{5\pi}{4}) + sin(\frac{5\pi}{4})$

This problem is original

Both are false Both are true (1) is true while (2) is false (2) is true while 1) is fakse

1 solution

Abhay Tiwari
Apr 19, 2016

Since:

$\sqrt{i}=\sqrt{e^{\frac{i\pi}{2}}}$

Then:

$\sqrt{i}=(+/-) {e^{\frac{i\pi}{4}}}$

$+ {e^{\frac{i\pi}{4}}} =cos(\frac{\pi}{4}) + sin(\frac{\pi}{4})$

and :

$- {e^{\frac{i\pi}{4}}} =-cos(\frac{\pi}{4}) - sin(\frac{\pi}{4})$

$-{e^{\frac{i\pi}{4}}} =cos(\frac{5\pi}{4}) + sin(\frac{5\pi}{4})$