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Derivatives of Sine and Cosine

I just checked Thomas's ninth edition of "Calculus".  As I expected, the derivative of the sine function took a couple pages of obscure arguments involving multiple lemmas, and I, at least, would have a hard time recreating it without a lot of head scratching after I'd looked away from it for more than about 10 seconds.  This seems to be typical of calculus texts -- the derivative of the sine function is treated as an important and deep fact, worthy of an important and difficult derivation.

In fact, the derivative can be found from a single picture.
We will show that later on this page (in section 4).  The picture is somewhat complex, however, so we're going to work up to it slowly, piecing it together from some slightly simpler diagrams which we'll present first.

1. Sine and Cosine of θ

Figure 1:  Definition of sin and cos (quadrant 1)

We will define the cosine and sine of θ as the x and y coordinates of a point on the unit circle whose radius forms that angle with the x axis.  This is shown for an angle in the first quadrant in figure 1.

We've shown (part of) a circle centered on the origin with radius = 1.
We've labeled the point where the radius line intersects the circle P.
The arc length of the segment of the circle stretching from the x axis up to point P is equal to θ (that is, in fact, the definition of the radian measure of an angle).
The length of the vertical line segment which runs from the x axis up to P is equal to sin θ.
The length of the horizontal line segment running from the y axis to P is equal to cos θ.

Figure 2: Magnified view showing sin and cos

In figure 2, we've shown an extremely magnified view of the region around point P in figure 1.

The segment of the circle which is shown in figure 2 has been magnified so much it appears almost flat.
The short jagged lines show where we've "cut off" the segments which run to the axes.
The segments marked "Radius", "Length = sin θ", and "Length = cos θ" are just as in figure 1.
We've marked the intersection of the radius line with the circle as being a right angle, and we've indicated a couple of angles in the figure as being equal to θ, which we can see by simple trigonometry.

2. Sine and Cosine of θ + δθ

Figure 3:  Sin and cos of θ+δθ

In figure 3, we've shown a magnified view, just as in figure 2, but this time it's for the sin and cos of angle (θ + δθ).
The magnification is so extreme, and the change in angle is so small, that the radius line may look like it's parallel to the radius line in figure 2, and has just been shifted up a centimeter or two.

3. δsin and δcos

Figure 4:  Difference in sin and cos when θ changes a little bit

In figure 4, we've shown the sine and cosine of θ along with the sine and cosine of θ+δθ, and we've indicated the differences between them in the figure.

The sine increased by the distance between the two horizontal gray lines.  That distance is indicated by "+δsinθ" in the figure.
The cosine decreased by the distance between the two vertical gray lines.  That distance is indicated as "-δcosθ" in the figure.
The segment of the circle running between the two radii has arc length "δθ", and is so marked in the figure.

4. Derivatives of Sine and Cosine in the First Quadrant

In figure 5 we've highlighted the little triangle formed by the segment of the circle between the two radii and the vertical and horizontal lines which we drew for sin θ and cos (θ+δθ).  It's a right triangle, with one angle equal to θ.  Its hypotenuse has length δθ.  Therefore its other two sides must have lengths sin θ δθ and cos θ δθ.

Figure 5: The proof

From figure 5, we can immediately read off the gist of the derivation:


To complete the derivation, we reduce the change in θ to an infinitesimal value, replace the δ characters with 'd' characters, divide through by , and fix up the signs, leading to:



Finally, to obtain what we might call the "defining property" of the derivative of the sine function, we use (4.3) to differentiate (4.2) again, leading to:


5. Other Quadrants

The visual derivation given above was done solely for angles in the first quadrant.
In fact, it works equally well for angles in any of the four quadrants, as long as one pays attention to the signs of the differences.  We won't show that, however, as the results of the other quadrants are easy to see just by looking at figure 5 and imagining the appropriate reflections.

6. If d2f/dx2 = -f, is f(x) necessarily a sine or cosine function?

Yes -- or, rather, it's a linear combination of sines and cosines.  The general solution to (4.4) looks like this:


We won't prove it, however.

Page created on 22 Sep 2013