Answer:
Assume that [tex]a[/tex] and [tex]p[/tex] are constants. The slope of the line will be equal to
- [tex]\displaystyle -\frac{\cos{(a)}}{\sin{(a)}} = \cot{(a)}[/tex] if [tex]\sin{a} \ne 0[/tex];
- Infinity if [tex]\sin{a} = 0[/tex].
Step-by-step explanation:
Rewrite the expression of the line to express [tex]y[/tex] in terms of [tex]x[/tex] and the constants.
Substract [tex]x\cdot \cos{(a)}[/tex] from both sides of the equation:
[tex]y \sin{(a)} = p - x\cos{(a)}[/tex].
In case [tex]\sin{a} \ne 0[/tex], divide both sides with [tex]\sin{a}[/tex]:
[tex]\displaystyle y = - \frac{\cos{(a)}}{\sin{(a)}}\cdot x+ \frac{p}{\sin{(a)}}[/tex].
Take the first derivative of both sides with respect to [tex]x[/tex]. [tex]\frac{p}{\sin{(a)}}[/tex] is a constant, so its first derivative will be zero.
[tex]\displaystyle \frac{dy}{dx} = - \frac{\cos{(a)}}{\sin{(a)}}[/tex].
[tex]\displaystyle \frac{dy}{dx}[/tex] is the slope of this line. The slope of this line is therefore
[tex]\displaystyle - \frac{\cos{(a)}}{\sin{(a)}} = -\cot{(a)}[/tex].
In case [tex]\sin{a} = 0[/tex], the equation of this line becomes:
[tex]y \sin{(a)} = p - x\cos{(a)}[/tex].
[tex]x\cos{(a)} = p[/tex].
[tex]\displaystyle x = \frac{p}{\cos{(a)}}[/tex],
which is the equation of a vertical line that goes through the point [tex]\displaystyle \left(0, \frac{p}{\cos{(a)}}\right)[/tex]. The slope of this line will be infinity.
