Why tangent show yoursel as slope line?

Question

The tangent is often drawn on graphs as slope line. Given that the tangent is a function that returns us a single number, I don't understand where the slope line comes from? If it shows the essence of the tangent, good. But then what is the formula for this tangent line?

Answer 1

The slope of the tangent line on a circle centered at the origin is equal to the negative of $\tan(\theta)$, where theta is the normal angle between the positive $x$-axis and the line from the origin to the point $P$.

The equation of the line itself can be found from a standard point-slope form: $$y-y_0=m(x-x_0),$$ where $m$ is the slope, found using calculus, this relationship, or possibly other methods.

Answer 2

It looks like you're being confused because you think of "tangent" as the name of a particular trigonometric function. But that is not the only meaning of that word.

In geometry, a "tangent" (or sometimes "tangent line") to a curve means a straight line that just touches the curve in question. That's exactly the line that's being drawn in this figure.

The trigonometric function is named for tangent lines: For angles between $0$ and $90^\circ$, $\tan(v)$ gives you the length of the segment of the tangent to the unit circle that lies between the "point of tangency" and the $x$-axis.

However, this naming doesn't mean that the tangent line has lost the right to be called a tangent. It had the name first!

Answer 3

Think of it in this way, tangent or specifically $\tan \theta$ is defined as ratio of perpendicular to the base, which means, it is the rate at which a point moves upwards (perpendicular to $x$-axis) in comparison to what moves horizontally (along the $y$-axis). That is what slope means actually.

When you get ready to climb a mountain, and you try to get a sense of its slope, the same process goes on your back of mind, most of the time it is unnoticed. When you have to move upwards a greater distance in comparison to what you go in the forward direction, you say its slope is steeper.

In calculus, you will learn how we associate each point on a graph with a tangent (if you know, then we take the first-order differentiation of the curve at that point) and essentially it is also equal to the trigonometric tangent of the angle that the line makes with the positive direction of $x$-axis.