Finding the equation of the straight line $y=ax+b$? 
If I have a circle $x^2+y^2=1$ and line that passes trough $(0,0)$
and I know the angle between the line and the axis. If, for example, the angle is $\frac{\pi}{3}$, how can I find the equation of the straight line $y=ax+b$?

What happens if angle is $\frac{\pi}{2}$? Best to use the general line equation of $ax+by+c=0$ instead of $y=\ldots$
So the general equation for a line with direction angle $\theta$ that passes through the point $(x_c,y_c)$ is
$$ (-\sin\theta)x + (\cos\theta)y + (x_c \sin\theta - y_c \cos\theta) = 0 $$
In your case $(x_c,y_c)=0$ so you have $$(-\sin\theta)x + (\cos\theta)y = 0 $$ or $$\boxed{ y = x \tan \theta } $$
