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 $y=ax+b$?

You do know that $b$ is called the "$y$-intercept", right?
There is a reason for that: $b$ is the value of $y$ where the line
crosses the $y$-axis.
Now look at the figure and see where the line crossed the axis and what
the value of $y$ is at that point.
Recall that $a$ is the slope of the line, which you can get from
the coordinates of two points on the line $(x_0,y_0)$, $(x_1,y_1)$
like so:
$$ a = \frac{y_1 - y_0}{x_1 - x_0}.$$
You already have one point, $(x_0,y_0) = (0,0)$.
So you just need to find one other point, for example
one of the points on the circle where the line intersects it,
or draw a right triangle with one leg from $(0,0)$ to $(1,0)$
(along the $x$-axis) and see where the third point is if the
angle at $(0,0)$ is $\frac\pi3$.
