Question

use identities to find values of the sine and cosine functions of the function for the angle measure.
2x, given \\( \tan x=-4 \\) and \\( \cos x<0 \\)
\\( \cos 2 x=-\frac{15}{17} \\)
(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.)

Explanation:

Step1: Determine the quadrant of \(x\)

Since \(\tan x=- 4<0\) and \(\cos x < 0\), \(x\) is in the second quadrant.

Step2: Find \(\sin x\) and \(\cos x\)

We know that \(\tan x=\frac{\sin x}{\cos x}=-4\), so \(\sin x=-4\cos x\).
Using the identity \(\sin^{2}x+\cos^{2}x = 1\), substitute \(\sin x=-4\cos x\) into it:
\((-4\cos x)^{2}+\cos^{2}x=1\)
\(16\cos^{2}x+\cos^{2}x = 1\)
\(17\cos^{2}x=1\)
\(\cos^{2}x=\frac{1}{17}\)
Since \(x\) is in the second quadrant, \(\cos x=-\frac{1}{\sqrt{17}}=-\frac{\sqrt{17}}{17}\)
Then \(\sin x=-4\cos x=-4\times(-\frac{\sqrt{17}}{17})=\frac{4\sqrt{17}}{17}\)

Step3: Use the double - angle formula for \(\sin2x\)

The double - angle formula for \(\sin2x\) is \(\sin2x = 2\sin x\cos x\)
Substitute \(\sin x=\frac{4\sqrt{17}}{17}\) and \(\cos x=-\frac{\sqrt{17}}{17}\) into the formula:
\(\sin2x=2\times\frac{4\sqrt{17}}{17}\times(-\frac{\sqrt{17}}{17})\)
\(\sin2x=2\times\frac{-4\times17}{17\times17}\)
\(\sin2x=-\frac{8}{17}\)

Answer:

\(\sin2x =-\frac{8}{17}\)