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**UPDATE:** I've created a new interactive trigonometry derivatives applet to replace the out-of-date one that was on this page.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