Find $\sin 2 x, \cos 2 x$, and $\tan 2 x$ if $\cos x=\frac{2}{\sqrt{5}}$ and $x$ terminates in quadrant IV. \[ \begin{array}{l} \sin 2 x= \\ \cos 2 x= \\ \tan 2 x= \end{array} \]
Solution
Step 1 ：We are given that \(\cos x = \frac{2}{\sqrt{5}}\) and that \(x\) is in the fourth quadrant. We can use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to find \(\sin x\). Since \(x\) is in the fourth quadrant, \(\sin x\) will be negative.
Step 2 ：Using the given value of \(\cos x\), we find that \(\sin x = -\frac{1}{\sqrt{5}}\).
Step 3 ：We can use the double angle formulas to find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\). The double angle formulas are: \(\sin 2x = 2 \sin x \cos x\), \(\cos 2x = \cos^2 x - \sin^2 x\), and \(\tan 2x = \frac{\sin 2x}{\cos 2x}\).
Step 4 ：Substituting the values of \(\sin x\) and \(\cos x\) into the double angle formulas, we find that \(\sin 2x = -0.8\), \(\cos 2x = 0.6\), and \(\tan 2x = -1.33\).
Step 5 ：So, the final answers are: \(\sin 2 x= \boxed{-0.8}\), \(\cos 2 x= \boxed{0.6}\), and \(\tan 2 x= \boxed{-1.33}\).
