FInd the sine and cosine of $2x$ given $\tan x = 3$ and $\sin x < 0$

Question

Find sine / cosine of $2x$, given $\tan x = 3$, $\sin x < 0$

The answer is $\cos 2x = -4/5$ and $\sin 2x = 3/5$

But why is $\cos 2x$ negative? What does $\sin x < 0$ mean?

Answer 1

$\tan x$ being positive tells you that $x$ is in the first or third quadrant, and $\sin x$ being negative tells you that $x$ is in the third or fourth quadrant. Conclusion: $x$ is in the third quadrant. This now implies that $2x$ is in the first or second quadrant.

Now, $\tan 2x = \frac{2 \tan x}{1 - \tan^2 x} = \frac{2(3)}{1 - 3^2} = -\frac{3}{4}$, you can imagine a $3$-$4$-$5$ triangle in the second quadrant with $\cos 2x = -\frac{4}{5}$ and $\sin 2x = \frac{3}{5}$.

$\cos 2x$ is negative because we're in the second quadrant. $\sin x < 0$ is just to specify which of the possible solutions it is.