<s>[INST] 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 ?!
 [/INST] You calculated your equations using the given $x$ (angles) in terms of degrees, not in radians: specifically, you need to multiply each angle in your equations of the lines $\,(75^\circ, 15^\circ\,)$ by the conversion factor $\frac \pi{180^\circ}$ to obtain those values in radians, which is how the solution equations are given.
Note that you've mixed degrees and radians when expressing the solutions to $x_1, x_2$:

$$x_{1}=75^\circ +k\pi$$ $$x_{2}=15^\circ+ k\pi$$ 

where you should express them as $$x_1 = \dfrac {5\pi}{12} + k\pi$$ $$x_2 = \dfrac{\pi}{12} + k\pi$$
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