How to solve for $~ 2x - \tan(x)=0~$ 
I need to find the roots for this function $$~ 2x - \tan(x)=0~$$ in order to graph it. 

I have found the one root $~(x=0)~$ but there are two more $~(x= -1.164 ,~ x= 1.164)~$. How can I find these answers without a graphing calculator ?
Or is this not possible without a calculator?
Thanks, any help is appreciated.
 A: Let $y=\frac{\pi}{2}-x$, to solve the complementary angle.  
$2x=\tan(x) → \frac{1}{\pi - 2y} = \tan(y)$
$$f(y) = \tan(y) - \frac{1}{\pi - 2y}$$
$$f'(y) = \sec(y)^2 - \frac{2}{(\pi-2y)^2}$$
Newton's method: $y_{i+1} = y_i - \frac{f(y_i)}{f'(y_i)}$, we got 0.5 → 0.407975 → 0.405237 → 0.405235   
$$x=\frac{\pi}{2}-y ≈ 1.16556$$
If trig. functions not allowed, interpolation work very well too.  
$f(\frac{\pi}{12}) = (2 - \sqrt 3) - \frac{1}{\pi-\pi/6} ≈ -0.1140227$
$f(\frac{\pi}{8}) = (\sqrt 2 - 1) - \frac{1}{\pi-\pi/4}  ≈ -0.0101996$
$f(\frac{\pi}{6}) = \frac{\sqrt 3}{3}-\frac{1}{\pi-\pi/3}≈ 0.0998854$
$f(\frac{\pi}{4}) = 1 - \frac{1}{\pi-\pi/2} ≈ 0.3633802$ 
$\begin{matrix}
f(y) & y \cr
-0.1140227 & \frac{\pi}{12} \cr
-0.0101996 & \frac{\pi}{8} & 0.405559 \cr
+0.0998854 & \frac{\pi}{6} & 0.401350 & 0.405169 \cr
+0.3633802 & \frac{\pi}{4} & 0.386855 & 0.405048 & 0.405215
\end{matrix}$
Inverse interpolation for $f(y)=0 → y ≈ 0.405215 → x ≈ 1.16558$ 
A: There is a far simpler method which does not involve calculus and which can reliably solve many equations of this kind. It does need a calculator, but not a graphing one. 
Consider the graphs of $\tan x$ and $2x$ at the point of intersection near $\pi$. The tangent curves upwards while the linear function is a straight line. 
In these circumstances, any estimate $x_0$ of the root of the equation will always be improved if you change it to $x_1=2\tan^{-1}x_0$. On the iPhone’s calculator this requires three keystrokes for each iteration. Repeating it moronically until the answer stops changing, one gets $$x=1.165561185207211$$
Whenever you have the right curvature of the intersecting graphs, this method will work. You need to decide which function to iterate, in this case whether $2\tan^{-1}x$ or $\frac12\tan x$, which you can do by sketching, by reasoning about curvature, or just trying it both ways and seeing which one converges and which one explodes.
Since the convergents are alternately too high and too low, you can accelerate a slow convergence by averaging the new value with the old one at each step. 
And yes, you can justify the soundness of this method with calculus. But the method itself is independent of it and comes into its own in cases where the functions concerned are hard or complicated to differentiate. 
A: Here is a closed form using “zeros of the Bessel function derivative” from DLMF defined by:
$$\frac{d\text J_n(x)}{dx}=\text J’_n(x)=\text J_{n-1}(x)-\frac{n\text J_n(x)}{x} =\frac{\text J_{n-1}(x)-\text J_{n+1}(x)}2=0\iff x\text J_{n-1}(x)=n\text J_n(x)\iff \text J_{n-1}(x)=\text J_{n+1}(x)\implies x=\text j’_{v,m}$$
where there appears the Bessel J function. Set $v=\frac12$:
$$\text J’_\frac12(x)=\frac{2x\cos(x)-\sin(x)}{\sqrt{2\pi}x^\frac32}=0\iff 2x\cos(x)=\sin(x)\iff \boxed{\tan(x)=2x\implies x=\text j’_{\frac12,m\in\Bbb N}}$$
where $m$ is the $m$th zero. This function is implemented in the DLMF, from the National Institute of Standards and Technology, so it may be a standard function. Please correct me and give me feedback!
A: This transcendental equation does not have a closed-form formula and if the goal is to plot, there is no real benefit in computing those roots.
What you can do instead, is to plot the function $\tan x$, which has a well-known shape and is periodic, and find the intersections with $y=2x$ just graphically.

In any case, the roots are close to and odd multiple of $\dfrac\pi2$.
