The Trigonometric Roundabout

How to differentiate and integrate cos(x) and sin(x).

I have recently found that several students struggle to deal with the differentiation and integration of the cos(x) and sin(x) functions. This usually occurs once the integral form of these functions is introduced, and students tend to get muddled as to where and when a sign change occurs.

Here is how you can construct a simple diagram that can assist you with the integration and differentiation of cos(x) and sin(x):

First of all write sin(x) above a -sin(x)

step 1 of diagram creation
Then complete a cross of terms by writing down -cos(x) and cos(x) from left to right, as shown:

cross of sin(x) and cos(x)

Now as we move clockwise through the terms we can find the derivative of the previous term.



Example 1

 y = sin(x) \rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}=cos(x)

Example 2

 y = -cos(x) \rightarrow \frac{\mathrm{d} y}{\mathrm{d} x}=sin(x)


Now the same diagram can be used in reverse (anticlockwise) to find the integral of the previous term.

Integrate Cycle

Example 3

 \int sin(x)=-cos(x)

Example 4

 \int -cos(x)=-sin(x)


I hope you’ll find the diagram as a helpful reminder for the differentiation and integration of sin(x) and cos(x), if so, please subscribe to my blog below (or by using the subscribe form in the side menu).

How to integrate and differentiate cos(x) and sin(x)


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