In this article we will show you how to derive the trigonometric functions (sine, cosine and tangent). You need these derivations for several topics, such as extreme points or turning points.
If you would like to get more information about the individual trigonometric functions, then take a look at the «Trigonometric functions» chapter. There you will find everything you need to know about these functions.
Derivation of trigonometric functions – overview
You can imagine the derivatives of the sine and cosine function as a kind of cycle. You can look at the following figure for this:
Figure 1: Deriving circle of sine and cosine function
If you remember this cycle, you have already understood an important part of the derivations.
You will find out how the derivation circle comes about in the next section.
You can also memorize this circle to form the antiderivative of the sine and cosine. All you have to do is run the arrows counterclockwise.
Derivation of the sine function
You already know the derivative of the sine function from the derivative circle. Let’s recap the whole thing mathematically:
If you want to learn how the derivative of the sine function comes about, you can look at the next in-depth section.
You can derive the derivative using the differential quotient.
So that you are well prepared for this, you should know the articles differential quotient and addition theorems.
The derivative is defined using the differential quotient as follows:
If you now insert the sine function, you get the following expression:
This is where you need to apply the sine addition theorem.
Addition theorem sine: .
Then you get the following:
Now you can first simplify this expression and apply the calculation rules for limit values:
Now you would have to form the limit value for both expressions. Since this would lead too far at this point, you simply have to believe the following two values:
This gives you the following derivative for the sine function:
Derivation of the cosine function
Thanks to the derivative circle, you know both the derivative of the sine function and the cosine function. You can now formulate this even more mathematically:
If you want to learn how the derivative of the cosine function comes about, you can look at the next in-depth section.
You can derive the derivative using the differential quotient.
So that you are well prepared for this, you should know the articles differential quotient and addition theorems.
The derivative is defined using the differential quotient as follows:
If you now insert the cosine function, you get the following expression:
This is where you need to apply the cosine addition theorem.
Addition theorem cosine: .
Then you get the following:
Now you can first simplify this expression and apply the calculation rules for limit values:
Now you would have to form the limit value for both expressions. Since this would lead too far at this point, you simply have to believe the following two values:
This gives you the following derivative for the cosine function:
Derivation of the tangent function
Unfortunately, the derivative circle says nothing about the derivative of the tangent function.
If you are wondering how the derivative of the tangent function comes about, you can look at the next in-depth section.
You can rewrite the tangent function as follows:
If you derive this function using the product rule, you get the following derivation:
You can also rearrange the equation as follows:
As a little reminder: .
This then results in the following derivation:
2(x)
So you have derived both derivatives.
Great, now you know all the derivatives of the pure trigonometric functions. Unfortunately, in many exercises you do not have the pure version of the trigonometric function, but with different parameters.
Derivatives of the extended trigonometric functions
More interesting are the derivatives of the extended trigonometric functions with the parameters. It might be helpful if you look again at our article on the derivation rules. In particular, you should have the chain rule ready!
Since you mainly need the derivatives of the sine and cosine function in school, only these two are considered here.
Determine the derivative of the extended sine function
You should calculate the derivative of the extended sine function.
To apply the chain rule, you first take the inner derivative of the function. Since the parameters are real numbers, the derivative of the function is as follows:
It will help you if you now rewrite the extended sine function:
You also need the derivative of the outer function. This corresponds to the sine function. All you have to do is derive the pure sine. Now you can look at the entire derivative of the extended sine function:
If you now use the functions and , you get the following derivation:
Well done, let’s apply the derivation you’ve learned to an example:
Task 1
Form the first derivative of the function with .
solution
First you need the inner derivative:
The sine function becomes the cosine function by deriving it, so you get the following solution:
Determine the derivative of the extended cosine function
You are to calculate the derivative of the extended cosine function.
To apply the chain rule, you first take the inner derivative of the function. The derivation of the function is as follows:
It can be helpful for you again if you rewrite the extended cosine function:
In addition, you need the derivative of the outer function again. This corresponds to the cosine function. All you have to do is derive the pure cosine. Now you can look again at the entire derivative of the extended cosine function:
If you now use the functions and , you get the following derivation:
Great, now you also know the derivative of the extended cosine function. Apply your newly acquired knowledge here as well:
exercise 2
Form the derivative of the function with .
solution
First you need the inner derivative:
The negative sine function is derived from the cosine function by deriving it. So you get the following first derivative:
Second and third derivatives of the extended trigonometric function
You need the second and third derivatives of the extended sine and cosine function for high and turning points. Since these form just like the first derivative, you don’t necessarily need to consider them separately. If you still want to look at them, you can look at the next in-depth section.
If the parameters for the sine and cosine functions are the same and there is an extreme point for the cosine function, then there is a zero for the sine function and vice versa.
Derivation of trigonometric functions – tabel
Finally, you can look at the following table as a summary:
You don’t have to memorize the derivations for the advanced functions. You can always apply the derivation rules to a given function. However, if you have problems with such derivations, you can also remember the derivations.
Derivation of trigonometric functions – exercises
In order to internalize the derivation rules a little more, you can consider the following task:
task 3
Calculate the first, second and third derivatives of the function with .
solution
You can now easily use the derivations from the table above, or you can derive the function yourself as an exercise. Here you can find the derivations with several steps.
Since you need the inner derivative for all derivatives, write them out first:
You can then form the first derivative as follows:
The second derivative is as follows:
You can then form the third derivative as follows:
You can now also look at an example using a sine function in order to internalize the derivations:
task 4
Calculate the first, second and third derivatives of the function with .
solution
You can decide again whether you use the derivations from the table or derive the function yourself.
Again, first write out the inner derivative:
The first derivative is as follows:
You can form the second derivative as follows:
You can calculate the third derivative as follows:
Derivative sine cosine tangent – The most important
- The derivatives of the trigonometric functions repeat themselves. You can remember this with the help of the derivation circle.
- The derivatives of the sine and cosine functions are as follows:
- You need the derivatives of the sine and cosine function for extreme and turning points.
- The derivative of the tangent function is: .