The definition of the slope of the line of the secant is:
slope = $\frac{y2-y1}{x2-x1}$
The definition of the slope of the tangent line is:
$\lim_{h->0}\frac{f(x+h)-f(x)}{h}$
I understand why they call it the tangent line since the angle to the x axis will be $tan(\theta) =\frac{Opp}{Adj}$ equivalent to opposite of adjacent.
Secant is the inverse trig function of cosine, so $\sec(\theta)=\frac{Hyp}{Adj}$
But I don't understand how secant is related to the slope of its line? I looked it up and I found out that the word secant comes from the Latin word secare, which means to cut. But is there any relation to secant and it's angle?
Best Answer
Because you can define $sec(\theta)$ as a length on the unit circle. $sec$ corresponds to the length of the line from $(0,0)$ to $(1, \tan(\theta))$ and $tan$ corresponds to the length of the segment from $(1,0)$ to $(1, \tan(\theta))$. See the figure here. Clearly the $sec$ segment cuts the circle and $tan$ is tangent to it.