Calculus-Free Derivatives of Sine and Cosine
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As a large square of side curls into a cylinder, an uncurled unit square is kept tangent to it, so that the unit square's diagonal always lies tangent to the helix formed by the larger square's diagonal. Projecting the helix into three mutually perpendicular planes yields a sinusoid, a "cosinusoid", and a circle; projecting the unit square's diagonal yields segments tangent to those curves. Basic geometric analysis of this figure then provides a straightforward development of the formulas for the derivatives of the sine and cosine functions from calculus… with nary a difference quotient to be seen!
Contributed by: B. D. S. Don McConnell (March 2011)
Open content licensed under CC BY-NC-SA
The controls in this Demonstration, taken in order, reveal aspects of a figure that helps "explain" (but not quite "prove" here) the calculus formulas for the derivatives of the sine and cosine functions. Describing those controls will facilitate discussion of the figure and the explanation.
The "displacement" slider adjusts the location of a unit square, whose diagonal remains aligned with that of a larger square. The value ranges from to and defines the horizontal and vertical displacement between the centers of the squares. (Geometric representations of these displacements appear in yellow when the "show displacements" checkbox is selected.) Think of the smaller square's diagonal as a vector—with unit-length horizontal and vertical components—oriented in the direction of increasing .
The "curvature" slider curls the larger square from a plane () into a cylinder of unit radius (), simultaneously twisting the larger square's diagonal into a helix. With the flat unit square kept tangent to the curling surface, the smaller orange diagonal always lies tangent to the evolving helix. (In this context, the "horizontal displacement" between the squares' centers becomes "angular displacement" of the smaller square's center about the axis of the cylinder.)
The "show projected curves" checkbox projects the helix in three mutually perpendicular directions, revealing a trio of familiar curves:
• The (blue) projection along the cylinder's axis is a unit circle.
• The (red) projection parallel to the original square is a sinusoid.
• The (green) last projection is also a sinusoid, but is better identified as a "cosinusoid".
The "show projected tangents" checkbox projects the unit square's diagonal vector (which is tangent to the helix) onto the planes, providing three vectors, with each vector oriented in the direction of increasing , and—most importantly—with each vector tangent to its corresponding projected curve. Thus, the projected vectors embody the "derivatives" of those curves.
To formalize further discussion, we introduce coordinates.
The "show axes" checkbox reveals a coordinate system in which the helix has parameterization . (The "negative" axes are dashed.) Under this coordinate system,
• The (blue) projected circle satisfies the relation .
• The (red) projected sinusoid satisfies the relation .
• The (green) projected "cosinusoid" satisfies the relation .
The coordinates will allow quantifying the "slopes" of the projected tangent vectors.
The "show slope triangles" checkbox decomposes each tangent vector into orthogonal components that represent the "slope" of each vector in its corresponding plane. (Note that the three vectors share a total of three orthogonal components, copies of which are drawn where the projection planes meet.) Components pointing in a "negative" coordinate direction are shown dashed.
• Let be the signed length of the (green) -directed component.
• Let be the signed length of the (red) -directed component.
• Let be the signed length of the (blue) -directed component. Note that is always 1.
The final control, the "show radial triangle" checkbox, shows a familiar right triangle within the (blue) projected circle. The hypotenuse—of length 1—is a radius inclined at angle from the positive axis, and thus meeting the projected tangent vector; the legs have (signed) lengths equal to (in green) and (in red). ("Negative" lengths are dashed.) The value of itself is represented as the (signed) area of a darkened sector of the circle.
Elementary geometry guarantees that a tangent to a circle is perpendicular to the radius that meets it. Consequently, the radial triangle is in fact congruent to the slope triangle of the circle's tangent segment: the hypotenuse of each of these triangles has length 1, with corresponding legs (marked with identical colors) merely aligned to different coordinate axes.
Observe that one pair of corresponding legs (the green pair) in these congruent triangles has both legs either solid or dashed; that is, their signed lengths agree in both length and sign. The other (red) pair of legs never match (unless they have length zero): one is dashed and one is solid; their signed lengths agree in length, but differ in sign. (Together with the fact that corresponding legs are aligned with different axes, this conforms to the expectation that the tangent segment and radius segment have "negative-reciprocal slopes".) Therefore,
so that we have quantified the final two of the three orthogonal components in the slope triangles for the projected vectors.
Using the elementary definition of the derivative of a function as the slope of the tangent vector to its graph, this recaptures the standard formulas from calculus:
All this without an explicit* appeal to limits of difference quotients, making the derivative formulas understandable without the sophisticated machinery (such as the Sandwich theorem) of differential calculus.
*Can you spot this explanation's implicit use of limits? Does this use invalidate the explanation as an independent-of-sophisticated-machinery proof of the derivative formulas?
See also: B.D.S. "Don" McConnell, "Calculus-Free Derivatives of Sine and Cosine".