Domain range of a straight line y = 2x + 3 for polar coordinate θ

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.

• 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? – Rory Daulton Feb 9 at 1:36

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.