Explore the slope of the sin curve
By Murray Bourne, 25 Oct 2010
UPDATE: I've created a new interactive trigonometry derivatives applet to replace the out-of-date one that was on this page.
The background below is still worth reading!
This was an important problem for mathematicians for centuries. If something is moving so that its velocity and acceleration are forever changing, how do we measure the rate of change?
It is easy to find rate of change (or slope, or gradient) for an object moving at constant speed, or constant acceleration, but what do we do if when the speed or acceleration is not constant, as is the case for objects moving in circular or regular paths (which we can describe using the trigonometric functions)?
This article, and the 2 that follow, shed some light on this interesting problem.
First, some reminders so you can better follow what is going on.
Tangent to a curve
A tangent to a curve means the line that touches the curve at one point only.
The tangent of the curve at the point A (screen shot)
(For more information, see Tangents and Normals).
The slope of a curve means the slope of the tangent at a particular point.
Here is the curve y = sin x. The values of x are in radians and one complete cycle goes from 0 to 2π (or around 6.28).
Graph of sin x
y = sin x
Our aim is to explore the slope of the curve y = sin x. Clearly. the slope of the curve changes as we move left to right.
Graph of cos x
Next we have y = cos x. It has the same shape as the sine curve, but has been displaced (shifted) to the left by π/2 (or 90°).
y = cos x
Slope of Sine x
First, have a look at the interactive graph below and observe that the slope of the (red) tangent line at the point A is the same as the y-value of the point B.
Then slowly drag the point A right or left and observe the curve traced out by point B. (The point B has the same x-value as point A, and its y-value is the same as the slope of the curve at point A).
Update: This applet has been replaced with a new one here:
Hopefully you can see that B traces out the curve y = cos x. In other words, the slope of the graph y = sin x at any point (x,y) has value cos x. Using calculus notation, we would write this as:
See more on the differentiation of sin, cos and tan curves. (This is in the calculus section of IntMath.)
See the 16 Comments below.