Find equation of a tangent on $y= \sin2x$ Find equation of a tangent on $y= \sin2x$ in intersections with $y=\frac{1}{2}$
What I calculated:
Intersections: $$\sin2x= \frac{1}{2}$$
...
$$0=tg^2x-4tgx-1$$
$$tgx_{1}=2+\sqrt{3}$$
$$x_{1}=75+k\pi$$ 
$$tgx_{2}=2-\sqrt{3}$$
$$x_{2}=15+ k\pi$$ 
$$T_{1}(75+k\pi,\frac{1}{2})$$ and $$T_{2}(15+k\pi,\frac{1}{2})$$
Derivative of $y= sin2x$ is: $y'= 2cos2x$
$$ y'(75)= -\sqrt {3} =k_{t_{1}} $$ 
$$y'(15)= \sqrt {3} =k_{t_{2}}$$ 
formula for calculating tangent that I used: $y-y_{1}=k_{t}(x-x_{1})$
1.)$T_{1}(75+k\pi,\frac{1}{2})$
$k_{t_{1}}=-\sqrt {3}$
$y_{1}-\frac{1}{2}=-\sqrt {3}(x-75-k\pi)$
$y_{1}=-\sqrt {3}x+75\sqrt {3}+\sqrt {3}k\pi+\frac{1}{2}$
2.)$T_{2}(15+k\pi,\frac{1}{2})$
$k_{t_{2}}=\sqrt {3}$
$y_{2}-\frac{1}{2}=\sqrt {3}(x-15 -k\pi)$
$y_{2}=\sqrt {3}x-15\sqrt {3}-\sqrt {3}k\pi+\frac{1}{2}$
Both solutions are wrong; what I should actually get:
$y_{1}=\sqrt{3}x-\frac{\sqrt{3}pi-6+12kpi\sqrt{3}}{12} $ and $y_{2}=-\sqrt{3}x-\frac{5\sqrt{3}pi+6+12kpi\sqrt{3}}{12};k=Z $
How could I possibly get that ?!

\begin{align*}
\ \sin{2x} &= \frac{1}{2} \\ 
\ \arcsin{\big(\sin{2x}}\big) &= \arcsin{\frac{1}{2}} \\ 
\ 2x &= \frac{\pi}{6} \\ 
\ x &= \frac{\pi}{12} \qquad\text{to be more precise you can say:}\\ 
\ x &= \frac{\pi}{12} + \pi k\quad ,\quad k\in\mathbb Z\\ 
\end{align*}
So we know $x=\dfrac{\pi}{12}$ when they intersect.
$y=a(x-x_0) + y_0$ is the formula for the tangent, where $a=f'(x_0)$.
So, $x_0 = \dfrac{\pi}{12} \land y_0=\dfrac{1}{2}$.
$f(x)=\sin{2x} \implies f'(x)=2\cos{2x}$
$f'(x_0) = f'(\dfrac{\pi}{12}) = 2\cos{\big(\dfrac{\pi}{6}\big)} = \sqrt{3} $
Thus, the tangent is $$y=\underline{\underline{\sqrt{3}\big(x-\frac{\pi}{12}\big) + \dfrac{1}{2}}}$$
