in which quadrant(s) are tan(x) and sin(x) both negative?\nquadrant ii\nquadrants ii and iii\nquadrant…

Explanation:

Step1: Recall sine - tangent signs in quadrants

Recall the signs of trigonometric functions in each quadrant. $\sin(x)=\frac{y}{r}$ and $\tan(x)=\frac{y}{x}$ where $(x,y)$ is a point on the terminal side of the angle $x$ and $r = \sqrt{x^{2}+y^{2}}>0$.

Step2: Analyze quadrant for negative sine

In quadrant I: $x>0,y>0$, so $\sin(x)=\frac{y}{r}>0$; in quadrant II: $x < 0,y>0$, so $\sin(x)=\frac{y}{r}>0$; in quadrant III: $x<0,y < 0$, so $\sin(x)=\frac{y}{r}<0$; in quadrant IV: $x>0,y < 0$, so $\sin(x)=\frac{y}{r}<0$.

Step3: Analyze quadrant for negative tangent

In quadrant I: $x>0,y>0$, so $\tan(x)=\frac{y}{x}>0$; in quadrant II: $x < 0,y>0$, so $\tan(x)=\frac{y}{x}<0$; in quadrant III: $x<0,y < 0$, so $\tan(x)=\frac{y}{x}>0$; in quadrant IV: $x>0,y < 0$, so $\tan(x)=\frac{y}{x}<0$.

Step4: Find common quadrants

We want the quadrants where both $\sin(x)<0$ and $\tan(x)<0$. From the above - analysis, the common quadrant is quadrant IV.

Answer:

C. quadrant IV