Is there a relationship between the trigonometric function tan(x) and the derivative of y with respect to x? Are they just named similarly by coincidence?
[Math] Relationship Between Tangent Function and Derivative
calculus
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Consider increasing the radius of a circle by an infinitesimally small amount, $dr$. This increases the area by an annulus (or ring) with inner radius $2 \pi r$ and outer radius $2\pi(r+dr)$. As this ring is extremely thin, we can imagine cutting the ring and then flattening it out to form a rectangle with width $2\pi r$ and height $dr$ (the side of length $2\pi(r+dr)$ is close enough to $2\pi r$ that we can ignore that). So the area gain is $2\pi r\cdot dr$ and to determine the rate of change with respect to $r$, we divide by $dr$ and so we get $2\pi r$. Please note that this is just an informative, intuitive explanation as opposed to a formal proof. The same reasoning works with a sphere, we just flatten it out to a rectangular prism instead.
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.
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.
Best Answer
Do you mean $\frac {dy}{dx}$ is reminiscent of the triangle definition of tangent $\tan \theta=\frac{\text{opposite}}{\text{adjacent}}$?
The derivative of a function at a point can be interpreted as the slope of the tangent line to that point on the graph of the function. This is distinct from the function tangent, which can be geometrically interpreted as the length of a special tangent to a unit circle (see below) given a certain angle.
You could connect them in a roundabout way - if you take the tangent line to a function's graph at a certain point, then extent it to intersect the x-axis, the tangent of the angle it forms with the x-axis (measured counterclockwise from the x-axis) will be the derivative of the function at that point.