Get the linear function without using tangent

Question

I have to get the linear function of a line on the rectangular coordinate system.

The Line I know that the line is $\ y=tan(90+\theta)\ x+1$ (degree, not RAD)

But is there a way to get the function without using tangent and(or) cotangent?

Answer 1

This question is equivalent to asking the question of what the side length of the side of the triangle in the picture on the x-axis is. If you know the equation of the line, you can calculate the y-intercept, which is the length of the side of the triangle on the x-axis, and if you know the side length of the triangle on the x-axis, you can compute the slope and use that formula to find the equation for the line. But the answer to the question "if I have an right triangle with acute angle $\theta$ and I want to find the ratio between the opposite and adjacent sides" is nothing more than a definition of what tangent is in the first place, so any solution not in terms of tangent or cotangent is just a derivation of how to express tangent.

More concretely, you might realize, for example, that by definition of cos, the length of the hypotenuse is just $\frac1{\cos(\theta)}$ and by definition of $\sin$, the length of the x-axis side is just $\sin(\theta)$ times the length of the hypotenuse, so putting this together, you get that the x-axis side of the triangle is $\frac{sin(\theta)}{\cos(\theta)}$, and hence, the slope of your line is $-\frac{\sin(\theta)}{\cos(\theta)}$, and thus, your line is given by $y = -\frac{\sin(\theta)}{\cos(\theta)} + 1$. But of course, this is just what you discovered in the first place since $\frac{\sin}{\cos} = \tan$.