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.)

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Answer

The final answer is $- \frac{4}{5}$ .

Steps

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 $\sin 2 x = 2 \sin x \cos x$ and $\cos 2 x = \cos^{2} x - \sin^{2} x$ to find the values of $\sin 2 x$ and $\cos 2 x$ .

Step 4 ：The value of $\cos 2 x$ is $- \frac{3}{5}$ .

Step 5 ：The value of $\sin 2 x$ is approximately -0.8.

Step 6 ：We need to simplify the expression for $\sin 2 x$ .

Step 7 ：The final answer is $- \frac{4}{5}$ .

Answer

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