# Differentiation Interactive Applet - trigonometric functions (sin, cos, tan, csc, sec and cot)

You can use this interactive applet to explore the derivatives of trigonometric functions as described in 1. Derivatives of the Sine, Cosine and Tangent Functions and the reciprocal functions, 2. Derivatives of Csc, Sec and Cot Functions .

Most of the time when we differentiate expressions involving trigonometric functions, we may not know what the original function looks like, or the nature of the derivatives we've actually found. This applet lets you see and explore the derivatives of simple trig functions.

### Things to Do

In this applet, there are pre-defined examples in the pull-down menu at the top.

In the left pane you will see the graph of the **function** of interest, and a triangle with base 1 unit, illustrating the slope of the tangent. In the right pane is the graph of the **first derivative** (the dotted curve).

- Select any of the examples in the pull-down menu.
- Use the
**slider**at the bottom to change the*x*-value. You can drag the slider left or right or you can animate the points by holding down the "−" or "+" buttons either side of the slider.

See below the graphs for some notes on each of the examples.

Choose function:

**Derivative:**

Slope

*dy*/*dx*

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## Some explanations

These fundamental trigonometric derivatives come from 1. Derivatives of the Sine, Cosine and Tangent Functions and 2. Derivatives of Csc, Sec and Cot Functions.

### Example 1

The first example is the sine function, `y=sin(x)`. The derivative curve is `dy/dx=cos(x)`.

### Example 2

The second example is the cosine function, `y=cos(x)`.

The derivative is `dy/dx=-sin(x)`, which is the reflection of `sin(x)` in the `x`-axis.

### Example 3

Next we have `y=tan(x)` which has periodic discontinuities at `... -(5pi)/2, -(3pi)/2,` `-pi/2, pi/2, (3pi)/2, ...`.

The derivative is `dy/dx=sec^2(x) = 1/(cos^2 x)`, which also has discontinuities at the same *x*-values as `y=tan(x)`.

### Example 4

Next up is `y=csc(x) = 1/sin(x)` which is has periodic discontinuities at `... -2pi, -pi,` `0, pi, 2pi, ...`.

The derivative is `dy/dx=-csc(x)cot(x)`, which has discontinuities at the same *x*-values as for `y=csc(x)`.

### Example 5

This one involves `y=sec(x)=1/cos(x)`, which has periodic discontinuities at `... -(5pi)/2, -(3pi)/2,` `-pi/2, pi/2, (3pi)/2, ...`.

The derivative is `dy/dx=sec(x)tan(x)`, which also has discontinuities at the same *x*-values as `y=sec(x)`.

### Example 6

The last one is `y=cot(x) = 1/tan(x)` which is has periodic discontinuities at `... -2pi, -pi,` `0, pi, 2pi, ...`.

The derivative is `dy/dx=-csc^2(x)`, which has discontinuities at the same *x*-values as for `y=cot(x)`.

**Credits:** Interactive based on a Java applet by David Eck and team from the Hobart and William Smith Colleges.

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