Is sine related to circle or curve?

Question

As tangent is a line passing through a curve in one point, and secant is a line passing through a curve in two points, is sine also something like this?

Answer 1

Tangent has two meanings:

1. The $\tan: \mathbb{R} \to \mathbb{R}$ function.

2. A straight line that intersects a curve at one and only one point.

The $\sin$ function represents the $y$-coordinate of a point on a circle.

Answer 2

In plane geometry, the words "secant" and "tangent" can be adjectives applied to a line that intersects a circle. A "tangent line" just barely touches the circle (hence tangent, from a Latin word for touching). A "secant line" cuts through the circle (hence secant, from a Latin word for cutting).

There are also (in trigonometry and calculus) functions named "secant" and "tangent". The secant of an angle $\theta$ is written $\sec(\theta)$. It is the length of the line segment indicated by the double-headed arrow next to which $\sec\theta$ is written in the figure below. The tangent of an angle $\theta$ is written $\tan(\theta)$. It is the length of the line segment indicated by the double-headed arrow next to which $\tan\theta$ is written in the figure below.

It so happens that the line segment labeled $\tan\theta$ in the figure is part of a tangent line of the circle, and the segment labeled $\sec\theta$ is part of a secant line of the circle. (The figure shows only one segment of the secant line, intersecting the circle once, but if you draw the rest of the line it will intersect the circle again in the lower left part of the figure.)

The sine function is also the length of a line segment in the figure; it is labeled $\sin\theta$. But there is no such thing as a "sine line" relative to the circle. In fact, if we draw the rest of the line on which the $\sin\theta$ segment lies, we can see that it's just another secant line.

In summary, "secant line" and "tangent line" describe entire lines in relatively generic ways; they also describe all possible ways in which lines can intersect a circle. The sine, tangent, and secant functions, in contrast, can be interpreted geometrically, but only as the lengths of specific line segments.