Connection between trigonometric identities and secant/tangent lines

Question

Assuming the relationship I am asking about is obvious to most students, I hope this post is an opportunity for some to have fun exploring a basic question. What I'm wondering about is the relationship to the trigonometric identities I learned about in PreCalculus and the secant/tangent lines that are used to estimate a rate of change at the start of Differential Calculus (or Calc I).

While I am can solve problems using the secant identity, $sec=\frac{r}{x}$, and I understand what it is (the inverse of cosine), I am having trouble connecting the relationship this identity has with the line I draw between two points on a curve, $m_{sec}=\frac{f(x)-f(a)}{x-a}$, also known as the difference quotient.

The same question comes up when I find the slope of a tangent line using the secant line. What is the relationship between the tangent I know from Trigonometry, $tan = \frac{y}{x}$, and the slope of the tangent line that I find in Calc I, $m_{tan} = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$?

I'm having trouble finding resources that address my questions directly online. So, any help would be greatly appreciated! I'll put in the time if you can point me in the right direction. Thank you!

Answer 1

The term "secant" and "tangent" have a more general meaning than just the trig function names.

A secant line is a line that intersects a curve in at least 2 distinct points.

A tangent line is a line that only "touches" a curve once.

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In the pre-calc definitions you give, the term $m_{\text{sec}}$ is called this way because it represents the slope of a line which intersects the curve given by the function $y=f(x)$ in 2 points: $(x,f(x))$ and $(a,f(a))$.

Similarly, the term $m_{\text{tan}}$ is called this way because it represents the slope of a line which only touches the curve at the point $(x,f(x))$.

In the diagram below you can visually see this. Here the red dot is the point $(x,f(x))$, the blue point is $(a,f(a))$, $m_{\text{sec}}$ would correspond to the slope of the purple line, and $m_{\text{tan}}$ corresponds to the slope of the orange line.

As far as the relation of these definitions of tangent and secant with how they're used in trigonometry, pyon's answer gives the diagram of the visual representation of secant and tangent functions as lines. Here we see that the secant function can be seen as a line that intersects the unit circle in 2 points, and similarly, the tangent function can be seen as another line which only touches the unit circle once.

Answer 2

The names of the trigonometric functions “tangent” and “secant” come from the following red triangle:

The circle above is the unit circle. The line that has length $\tan \theta$ is tangent to the circle, whereas the line that has length $\sec \theta$ is secant to the circle.

As far as I can tell, there is no relation whatsoever between, on the one hand, the functions $\tan$ and $\sec$, and on the other hand, tangent and secant lines to arbitrary curves.