Questions: Use identities to find values of the sine and cosine functions of the function for the angle measure. 2 x, given tan x=-4 and cos x<0 cos 2 x= (Simplify your answer, including any radicals. Use integers or fractions for any numbers the expression.)

Solution Steps

To find the values of the sine and cosine functions for the angle measure \(2x\), given \(\tan x = -4\) and \(\cos x < 0\), we can use trigonometric identities. First, determine the quadrant in which angle \(x\) lies. Since \(\tan x\) is negative and \(\cos x\) is negative, \(x\) is in the second quadrant. Use the identity \(\tan x = \frac{\sin x}{\cos x}\) to express \(\sin x\) and \(\cos x\) in terms of a common variable. Then, use the double angle identities: \(\cos 2x = \cos^2 x - \sin^2 x\) or \(\cos 2x = 1 - 2\sin^2 x\) to find \(\cos 2x\).

Given \(\tan x = -4\), we can express \(\sin x\) and \(\cos x\) in terms of a variable \(k\): \[ \sin x = -4k \quad \text{and} \quad \cos x = k \] Using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), we have: \[ (-4k)^2 + k^2 = 1 \implies 16k^2 + k^2 = 1 \implies 17k^2 = 1 \implies k = \frac{1}{\sqrt{17}} \] Thus, we find: \[ \cos x = -\frac{1}{\sqrt{17}} \quad \text{and} \quad \sin x = -\frac{4}{\sqrt{17}} \]

Using the double angle identity for cosine: \[ \cos 2x = \cos^2 x - \sin^2 x \] Substituting the values of \(\sin x\) and \(\cos x\): \[ \cos 2x = \left(-\frac{1}{\sqrt{17}}\right)^2 - \left(-\frac{4}{\sqrt{17}}\right)^2 \] Calculating each term: \[ \cos 2x = \frac{1}{17} - \frac{16}{17} = -\frac{15}{17} \]

Final Answer