tangents of a sine wave trough

These are the tangent lines defined by the trough of a sine wave, with x between -pi and 0. Over this particular stretch, the curve might be hard to distinguish from the parabola from x varying from -1 to 1, but a close look at the tangent patterns should show how the two curves are different. Notice where the first downward sloping red tangent line meets the last upward sloping purple line.

Here in the sine wave curve, the first few tangents on the left are almost on top of each other, as are the last few tangents on the right.

Here are the parabola tangents again. The slopes of these tangents are increasing steadily, while the sine curve tangents max out at a slope of 1 and begin to get smaller as we leave the trough.

Tomorrow, another famous "round valley" curve, the bottom half of a circle.