# Explore the slope of the tan curve

By Murray Bourne, 27 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**UPDATE:** I've created a new interactive trigonometry derivatives applet to replace the out-of-date one that was on this page.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 is the 3rd article in a series. The first 2 were: Explore the slope of the sin curve and Explore the slope of the cosine 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. Now, we are going to use the word "tangent" in 2 different, but related ways.

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

A **tangent** is the value of the fraction **opposite / adjacent** in a right triangle. (See Sine, Cosine and Tangent.)

## Curve Reminders

#### Graph of tan *x*

Below is the curve *y = *tan *x*, in blue. Recall that

Since cos *x* has value 0 when *x* = ..., -π/2, π/2, 3π/2, 5π/2, ... then our curve has **discontinuities **("holes") for these *x*-values, since division by 0 gives an undefined result.

This means we'll have an **asymptote** at those values. An asymptote is a line the curve gets closer and closer to, but does not touch or intersect, shown as a red dashed line below.

*y = *tan *x*

**Note: **The slope of the tangent curve is **positive** for all values of *x* (except the asymptotes, of course, where the curve and the slope is undefined).

#### 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*

#### Graph of sec *x* and sec^{2}*x*

We'll also need the **reciprocal** of the cosine curve, which we recall is defined as the **secant**.

To obtain the graph of *y* = sec *x*, we just take the reciprocal of each *y* -value of the cosine curve. So for example the point (1.05 , 0.5) (shown as a gray dot) of the cosine curve becomes (1.05, 2), (shown as a blue dot) of the secant curve.

In this graph, the cosine curve is shown in gray, and the secant curve in blue.

*y = *cos *x * (in gray) and *y* = sec *x* (in blue)

One more step - we need the **square** of the secant graph values. that is, we are going to graph *y* = sec^{2}*x**.* To do that, we just take each *y*-value of the secant curve and square it.

So for example the point (1.05, 2) on the *y* = sec *x* curve becomes (1.05, 4) on the *y* = sec^{2}*x* curve , as shown in the graph below.

The graph of *y* = sec *x* is shown in light blue, while the graph of *y* = sec^{2}*x** *is shown in dark blue. Of course, all the *y*-values of our new graph are positive, a result of squaring.

*y* = sec *x* (in light blue) and *y* = sec^{2}*x* (in dark blue)

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

## Slope of tan *x*

First, have a look at the graph below and observe 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 **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 *y* = sec^{2}*x*. In other words, the slope of the graph *y* = tan *x* has value sec^{2}*x*. Notice the slope is positive for all values of *x*.

Using calculus, we would write this result as:

See the other 2 articles in this series:

Investigate the slope of *y* = sin *x*

Investigate the slope of *y* = cos *x *

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

See the 2 Comments below.