The circle's arc leaves a question in the air: I know sine as a wave — what does it have to do with this circle? The answer is literal, not analogical: the height of the point is the height of the graph, and only the input has changed places. In the circle, it is a rotation; in the graph, a position.
Do: drag a and observe the dashed line — it is horizontal on purpose, because the two halves of the screen share the same vertical scale and the same zero line. Press animate to see the curve being drawn, and turn on show tangent and show arc to compare the three lengths.
1. The dashed line is horizontal. It is not decoration: it is the assertion that the height of the point on the circle is the height of the graph. The circle does not become a graph by analogy — it becomes one because the output is the same and only the input changes places. In the circle, the input a is a rotation; in the graph, it is a horizontal position.
2. The dashed vertical line is the tangent line to the circle at (1,0). That's why the tangent is called tangent: the value tan a is a length measured
along the tangent line. The point T is where the line through the origin and P
intersects that line — and that's why T disappears when P approaches the vertical: the two
lines become parallel and no longer intersect. The asymptote of the tangent graph is that,
seen up close.
3. The arc has length |a|. This is the definition of radian, and it is what allows us to compare the three: height < arc < tangent. This inequality, obtained without any calculation, is what proves the limit of sin(x)/x.
The word tangent names two objects: the length along the tangent line to the circle, which is what this page draws, and the tangent line to a curve — the one that touches and follows the slope. They are not interchangeable, but they are also not mere homonyms: they meet in a single identity.
f ′(x) = tg θ
The derivative is the trigonometric tangent of the angle θ that the tangent line makes with the x-axis. Slope of line is tangent of angle — the same ratio of the triangle, with the triangle lying on the curve instead of inside the circle.
Local, self-contained instrument: no network, no external library. Companion of Station 5 of C1 · Read a graph.