Jtdylktuy

This is the "symmetric form" of a line:

$$frac{x-x_1}{sintheta}=frac{y-y_1}{costheta}$$
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?

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 $m = tantheta$, where $theta$ is the line's angle on inclination, and since $tantheta = sintheta/costheta$, the symmetric form
$$frac{x-x_1}{costheta} = frac{y-y_1}{sintheta} tag{2}$$
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 $costheta$) from vertical "rise" components ($y-y_1$ and $sintheta$). 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 $(costheta,sintheta)$ 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

$$x = x_1 + tcostheta qquad y = y_1 + tsintheta tag{4}$$
or, more compactly,
$$(x,y) = (x_1,y_1) + t,(costheta,sintheta) tag{5}$$
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 $v=(costheta,sintheta)$. 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.)