Problem
Find $\sin 2 x, \cos 2 x$, and $\tan 2 x$ if $\tan x=\frac{3}{2}$ and $x$ terminates in quadrant $I$. \[ \sin 2 x= \] \[ \cos 2 x= \] \[ \tan 2 x= \]
Solution
Step 1 ：Given that \(\tan x = \frac{3}{2}\) and \(x\) is in the first quadrant, we can use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to find \(\sin x\) and \(\cos x\).
Step 2 ：Using the given value of \(\tan x\), we can find \(\sin x = 0.832\) and \(\cos x = 0.555\).
Step 3 ：We can then use the double angle formulas to find \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\).
Step 4 ：Using the double angle formulas, we find that \(\sin 2x = 0.923\), \(\cos 2x = -0.385\), and \(\tan 2x = -2.4\).
Step 5 ：Final Answer: \(\sin 2 x = \boxed{0.923}\), \(\cos 2 x = \boxed{-0.385}\), \(\tan 2 x = \boxed{-2.4}\)
