Find $\sin (2 x), \cos (2 x)$, and $\tan (2 x)$ from the given information. \[ \begin{array}{l} \tan (x)=-\frac{1}{6}, \cos (x)>0 \\ \sin (2 x)=\square \\ \cos (2 x)=\square \\ \tan (2 x)=\square \end{array} \] Need Help? Read It Watch it
Solution
Step 1 ：We are given that \(\tan(x) = -\frac{1}{6}\) and \(\cos(x) > 0\).
Step 2 ：We need to find \(\sin(2x)\), \(\cos(2x)\), and \(\tan(2x)\).
Step 3 ：We know that \(\sin(2x) = 2\sin(x)\cos(x)\), \(\cos(2x) = \cos^2(x) - \sin^2(x)\), and \(\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\).
Step 4 ：We can find \(\sin(x)\) and \(\cos(x)\) from the given \(\tan(x)\) using the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\) and the fact that \(\tan(x) = \frac{\sin(x)}{\cos(x)}\).
Step 5 ：Since \(\cos(x) > 0\) and \(\tan(x) < 0\), we know that \(\sin(x) < 0\) (because the tangent of an angle is negative if and only if the sine and cosine have opposite signs).
Step 6 ：So, we can find \(\sin(x)\) and \(\cos(x)\), and then substitute these values into the formulas for \(\sin(2x)\), \(\cos(2x)\), and \(\tan(2x)\) to find the answers.
Step 7 ：Final Answer: \[\begin{array}{l} \sin (2 x)=\boxed{-0.324} \\ \cos (2 x)=\boxed{0.946} \\ \tan (2 x)=\boxed{-0.343} \end{array}\]
