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A straight line is defined by equation $y=2x+3$ in Cartesian coordinate system XY. I need to specify the domain range for the polar coordinate u which is valid for this straight line.

Firstly, to convert to polar coordinates - $y = r \sin θ$ and $x = r \cos θ$

So from $r \sin θ = 2 * r \cos θ + 3$ which can be rewritten as $f(θ) = 3 / (\sin θ - 2 \cos θ)$

I'm aware that the domain range is determined by the slope. If $y = mx$, the slope is $m$. Hence, the domain range is determined by the line slope which is $2$.

My main issue is that I don't understand is how the range is derived

$\arctan(2) < θ < \arctan(2) + π$

For instance, why is $\arctan(2) + π$ allowed? I'm having trouble visualizing this.

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    $\begingroup$ Are you asking for a rigorous demonstration that the correct range for $\theta$ is given by those inequalities or a visualization of them or some other intuitive explanation of them? $\endgroup$ – Rory Daulton Feb 9 at 1:36
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Here's a graph of the line $y = 2x + 3$ along with some rays outward from the origin. Each ray is labeled with the angle of the ray in polar coordinates.

The rays labeled $\theta_1$ and $\theta_2$ lie along the line through the origin parallel to the line $y = 2x + 3$. This line has slope $2$ and the angle of the ray $\theta_1$ is therefore $\theta_1 = \arctan(2).$

The ray $\theta_2$ is in the exact opposite direction from $\theta_1.$ To rotate the ray $\theta_1$ counterclockwise onto $\theta_2,$ we have to rotate it $180$ degrees, which is $\pi$ radians. Therefore $\theta_2 = \theta_1 + \pi = \arctan(2) + \pi.$

As we rotate a ray so that its angle increases from $\theta_1$ to $\theta_2,$ at each angle (for example $\theta_3,$ $\theta_4,$ and $\theta_5$), the ray intersects the line and we can use that angle $\theta$ and the radius $r = f(\theta) = 3 / (\sin \theta - 2 \cos \theta)$ to plot one of the points on the line. As the ray sweeps through all the angles between $\theta_1$ and $\theta_2,$ the polar coordinates $(f(\theta), \theta)$ plot every point on the line.

We want all of those angles to be in the domain of $f$ so that we can plot the whole line. If you stopped at some angle before $\arctan(2) + \pi,$ for example $\theta_4,$ you would only have plotted part of the line. All the parts of the line at angles between $\theta_4$ and $\arctan(2) + \pi$ would be missing.

Note that $\arctan(2) + \pi$ actually is not "allowed," because the upper bound of the domain is described by $\theta < \arctan(2) + \pi$; notice how it is $<$ rather than $\leq,$ which tells us that the angle $\theta$ can never exactly equal $\arctan(2) + \pi.$ We cannot include either $\arctan(2)$ or $\arctan(2) + \pi$ in the domain, because the rays at those angles are parallel to the line and never intersect it, so there is no value we can set $r = f(\theta)$ to in order for $(f(\theta), \theta)$ to be polar coordinates of a point on the line.

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