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Explore the slope of the cos curve

By Murray Bourne, 26 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UPDATE: I've created a new interactive trigonometry derivatives applet to replace the out-of-date one that was on this page.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 article is the second in a series. (Here's the first installment: Explore the slope of the sin curve, and the 3rd: Explore the slope of the tan curve.)

First, some reminders so you can better follow what is going on.

The slope of a curve means the slope of the tangent at a particular point. A tangent to a curve means the line that touches the curve at one point only. (See Tangents and Normals).

Curve Reminders

Graph of cos x

Here is the curve y = cos x. The values of x are in radians and one complete cycle goes from 0 to (or around 6.28).

cos graph
y = cos x

Graph of sin x

The graph of y = sin x has the same shape as the cosine curve, but has been displaced (shifted) to the right by π/2 (or 90°).

sine graph
y = sin x

We now graph the negative of the above curve, that is, y = −sin x. To achieve this, we reflect the curve in the x-axis. This has the effect of multiplying each y-value in the curve by −1. Putting this another way, we "turn the graph upside-down", through the x-axis.

y - -sin x
y = −sin x

Now we are ready to investigate tne slope of the curve y = cos x using a GeoGebra-based JSXGraph interactive graph.

First, have a look at the graph below and observe the slope (m = -0.5) of the (red) tangent line at the point A is the same as the y-value of the point B (0.53, -0.5).

Then slowly drag the point A and observe the curve traced out by 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:

Interactive trigonometry derivatives applet

Hopefully you can see that B traces out the curve −sin x. In other words, the slope of the graph y = cos x at any point (x,y) has value −sin x. Using calculus, we would write this as:


See the first article in this series: Investigate the slope of y = sin x

In the next article, we'll examine the slope of y = tan x.

See more on the differentiation of sin, cos and tan curves. (This is in the calculus section of IntMath.)

See the 3 Comments below.

3 Comments on “Explore the slope of the cos curve”

  1. Tim Bates says:

    Hi Murray,
    Nice article and graphics!
    I cam seeking to learn what the maximum velocity of an object tracing a sinusoid is (in fact, what the maximum velocity of a piston is, given it is driven sinusoidally, and we know its RPM.

    Any chance you can link to such an article, or place the sin cos relationship into an example in which we can compute the maximum velocity a certain number of Hz entails?


  2. Murray says:

    @Tim: The maximum velocity of a piston occurs when it is in the exact middle of its travel.

    For an example, if the piston is operating at 5 Hz and its amplitude is 6 cm, then a model for its position (height) at time t is given by:

    h = 6 sin 10πt

    The piston will be at the middle of its travel when t = 0, or 2.5 and then again at 5, etc.

    The derivative is given by

    v = dh/dt = 60π cos 10πt

    At t = 0, the velocity is 60π ≈ 188.5 cm/s.

    I've done it from a calculus perspective. Here's another approach:

  3. Tim Bates says:

    Perfect. Thank you!

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