The points of contact of tangents to the circle $x^2 + y^2 = 25$ which are inclined at an angle of $30°$ to the $x-$axis are

The equation of a tangent of slope $m$ to the circle $x^2+y^2=a^2$ is $y=mx\pm a\sqrt{1+m^2}$ and the coordinates of point of contact are 

$\left(\mp \frac{am}{\sqrt{1+m^2}},\pm \frac{a}{\sqrt{1+m^2}}\right)$ 

Here, $a=5$ and $m=\tan 30=1/\sqrt{3}.$

So, the coordinates of the points of contact are $\left(\mp \frac{5}{2},\pm \frac{5\sqrt{3}}{2}\right)$