# 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**UPDATE:** I've created a new interactive trigonometry derivatives applet to replace the out-of-date one that was on this page.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!

## Background

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 2π (or around 6.28).

*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°).

*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*

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.

21 Nov 2015 at 3:09 am [Comment permalink]

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?

tim

23 Nov 2015 at 11:48 am [Comment permalink]

@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

tis given by:h= 6 sin 10πtThe 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 10πtAt

t= 0, the velocity is 60π ≈ 188.5 cm/s.I've done it from a calculus perspective. Here's another approach:

http://farside.ph.utexas.edu/teaching/301/lectures/node143.html

24 Nov 2015 at 4:08 am [Comment permalink]

Perfect. Thank you!