The graph of \(f(x)=5+0.4\tan (3x-0.5)\) completes six cycles between \(0\) and \(2\pi\text{,}\) so we expect to find six solutions, as illustrated below. We’ll use the substitution \(\theta=3x-0.5\) to reduce the equation to \(5+0.4\tan (\theta)=4.5\text{.}\) Next, we isolate the trig ratio.

Subtract 5 from both sides of the equation, then divide by 0.4.

\begin{align*}
0.4\tan \theta \amp = -0.5\\
\tan \theta \amp = -1.25
\end{align*}

Now we can solve for \(\theta\text{.}\)

\begin{equation*}
\theta = \tan^{-1}(-1.25)=-0.8960
\end{equation*}

Replacing \(\theta\) by \(3x-0.5\text{,}\) we find the first solution.

\begin{align*}
3x-0.5 \amp = -0.8960\\
3x\amp = -0.3960\\
x \amp = -0.1320
\end{align*}

This value of \(x\) is not between \(0\) and \(2\pi\text{,}\) but because the period of \(f(x)\) is \(\dfrac{\pi}{3}\text{,}\) we can add \(\dfrac{\pi}{3} \approx 1.0472\) to any solution to find another solution.

\begin{align*}
x_{1} \amp =-0.1320+1.0472=0.9152\\
x_{2} \amp =0.9152+1.0472=1.9624\\
x_{3} \amp =1.9624+1.0472=3.0096\\
x_{4} \amp =3.0096+1.0472=4.0568\\
x_{5} \amp =4.0568+1.0472=5.1040\\
x_{6} \amp =5.1040+1.0472=6.1512
\end{align*}

We stop here, because the next solution is greater than \(2\pi\text{.}\) Rounded to two decimal places, the six solutions are 0.92, 1.96, 3.01, 4.06, 5.10, and 6.15.