Use identities to find values of the sine and cosine functions of the function for the angle measure.
$2 x$, given $\tan x=-2$ and $\cos x< 0$
\[
\cos 2 x=-\frac{3}{5}
\]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
\[
\sin 2 x=
\]
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Answer
The final answer is \(\boxed{-\frac{4}{5}}\).
Step 1 ：We are given that \(\tan x = -2\) and \(\cos x < 0\).
Step 2 ：We can use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to find the value of \(\sin x\) and \(\cos x\).
Step 3 ：Then we can use the double angle formulas \(\sin 2x = 2\sin x \cos x\) and \(\cos 2x = \cos^2 x - \sin^2 x\) to find the values of \(\sin 2x\) and \(\cos 2x\).
Step 4 ：The value of \(\cos 2x\) is \(-\frac{3}{5}\).
Step 5 ：The value of \(\sin 2x\) is approximately -0.8.
Step 6 ：We need to simplify the expression for \(\sin 2x\).
Step 7 ：The final answer is \(\boxed{-\frac{4}{5}}\).
