Your friend was "half-way" right:
$(1)$ First finishing on your work, for $\,\tan\theta = \left(-\dfrac{3}{5}\right),\;$ if $\,\sin\theta < 0,\,$ we'd also need
$$\cos\theta = \dfrac{5}{\sqrt{34}} = \dfrac{5 \sqrt{34}}{34} > 0.\;$$ This makes sense since
$$\;\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{\frac{-3\sqrt{34}}{34}}{\frac{5\sqrt{34}}{34}} = \left(-\dfrac 35\right). $$
This, together with your work, gives that $P$ would then be in the fourth quadrant.
$(2)$ But... $P$ could also be located in Quadrant II: Since $\;\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \left(-\dfrac 35\right)\lt 0,\;$ then exactly one of $\cos\theta, \sin \theta\,$ must be negative. $(1)$ gives one possible way this can happen.
But we might also have that $\cos \theta \lt 0, \sin\theta > 0$, putting $P$ in Quadrant II.
$$\;\tan\theta = \dfrac{\sin\theta}{\cos\theta} = \dfrac{\frac{3\sqrt{34}}{34}}{\frac{-5\sqrt{34}}{34}} = \left(-\dfrac 35\right).$$
Recall that $\,\sin\theta\,$ corresponds to the $\,y$-coordinate of $\,P\,$ in a unit circle, and $\,\cos \theta\,$ with its $\,x$-coordinate. When $\,x > 0\,$ and $\,y< 0,\,$ $\,(x, y)\,$ is in the Fourth Quadrant; when $\,x \lt 0,\;\text{and}\; y\gt 0,\;$ $(x, y)\,$ is in the Second Quadrant.
