I am stuck in this complex number problem.
Let $z = -1 + 1i$. The polar form is $r = \sqrt 2$, $\tan (\theta) = 1/-1$.
My question is that i cannot determine which quadrant this lies.
I saw an answer that $\theta = \tan^{-1}(1/(-1))$, which lies in 4th quadrant.
But the answer to this is $(3\pi)/4$.
My question is how can I get that this value lies in $Q2$.
What I understand is that since $\tan \alpha =\sin\alpha /\cos\alpha$.
Here sine is positive and cosine is negative so this should be $Q2$ and the angle from triangle is $\pi/4$ which is rhe reflection of $3\pi/4$.
Many thanks. Still confuse at this.
Best Answer
Let $z=-1+i=x+yi,$ with $x=-1$ and $y=1$.
Clearly $z$ is in the second quadrant, since $x<0$ and $y>0$.
The polar form is $z=r(\cos\theta+i\sin\theta)$.
As you noted, $r=\sqrt2$.
Therefore, $\cos\theta=-\dfrac1{\sqrt2}$ and $\sin\theta=\dfrac1{\sqrt2}$,
so we can take $\theta=\dfrac{3\pi}4$.
Note that $\dfrac{\pi}2<\theta<\pi$, so $\theta$ is indeed in the second quadrant.