This is the "symmetric form" of a line:
$$\frac{x-x_1}{\sin\theta}=\frac{y-y_1}{\cos\theta}$$ where $\theta$ is inclination of the line.
I know the equation, but it seems to me like the point-slope form of the line. Can you please tell me why it's called a symmetric form?
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$\begingroup$This is very much like the point-slope form of the line, which could be written$$\frac{y-y_1}{x-x_1}=m \tag{1}$$where $m$ is the line's slope. But, since $m = \tan\theta$, where $\theta$ is the line's angle on inclination, and since $\tan\theta = \sin\theta/\cos\theta$, the symmetric form$$\frac{x-x_1}{\cos\theta} = \frac{y-y_1}{\sin\theta} \tag{2}$$is (mostly) equivalent to point-slope form via cross-multiplication.
Separating the $x$ and $y$ elements "symmetrizes" the equation by isolating the line's horizontal "run" components ($x-x_1$ and $\cos\theta$) from vertical "rise" components ($y-y_1$ and $\sin\theta$). The form is also symmetric in that now both horizontal ($\theta = 0^\circ$) as well as vertical ($\theta=90^\circ$) lines are problematic, since both directions lead to vanishing denominators. (That's why I used wrote "(mostly) equivalent" above: point-slope form is only problematic in one direction.)
It may seem a bit of form-over-substance here, but if we recognize $(\cos\theta,\sin\theta)$ as the line's "direction vector", then this form generalizes nicely to higher dimensions. For instance, in $3$D, we have
$$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c} \tag{3}$$
where $(a,b,c)$ is a direction vector of the line (with "run", "rise", and ... um ... "rother" components).
By the way, since the fractions in $(2)$ (and $(3)$) are equal to each other, they're equal to a common ratio that we can call, say, $t$. That means we can re-express line equations by writing
$$x = x_1 + t\cos\theta \qquad y = y_1 + t\sin\theta \tag{4}$$or, more compactly,$$(x,y) = (x_1,y_1) + t\,(\cos\theta,\sin\theta) \tag{5}$$or, even-more-compactly,$$p = p_1 + t\,v \tag{6}$$This is the parametric form of the line ($t$ is the "parameter"). Here, too, there's conceptual separated, since the calculations are done component-wise, but we get to see those components in meaningful groups: variable point $p=(x,y)$, given point $p_1=(x_1,y_1)$, direction vector $v=(\cos\theta,\sin\theta)$. This is effectively a "flattened", ratio-free version of the symmetric form.
It may be safe to say that one is far more likely to see $(5)$ or $(6)$ than $(2)$, since the non-ratio forms cleverly avoid the zero-denominator problem by not having denominators at all. (You might see $(2)$ or $(3)$ in a context where an author doesn't want to bother wasting a variable on an unused parameter.)
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