# Explainer: Differentiation of Trigonometric Functions

In this explainer, we will learn how to find the derivatives of trigonometric functions and how to apply the differentiation rules on them.

Having learned how to differentiate polynomial functions, we would like to extend our knowledge to the derivatives of trigonometric functions.

We will begin by considering the sine function and what we can learn about its derivative by considering its graph. Recall that we can extend sine to be a function defined on all real numbers and we write where is measured in radians. We will begin by considering the graph of the sine function.

We can now try to sketch a graph of its derivative. Notice that the turning points are at for —this is where the derivative will be zero. Furthermore, at the point where the sine graph meets the -axis, its derivative is at a maximum or minimum. In particular, when for , the slope of its tangent line is 1 and when for , the slope of the tangent line is . Hence, the graph of the derivative of sine is as follows.

An astute eye will notice that the graph of the derivative of sine appears to be the same as the graph of the cosine function. This is in fact true, and we will use the next example to prove this fact. However, the technique of considering the graphs of functions and sketching their derivatives can help us gain insight about the nature of its derivate. Before we formally differentiate sine, we recap the definition of the derivative of a function.

### Derivative

The derivative of a function is defined as at the points where the limit exists.

### Example 1: Differentiating Sine from First Principles

Differentiate from first principles.

Recall that the derivative of a function is defined by the limit

Substituting in sine for , we have

To simplify the expression within the limit, we can use our knowledge of trigonometric identities. In particular, we can use the sum formula, to rewrite this expression as

Using the sum and product rules of limits, we can rewrite this as

Since both and are independent of , their limits will simply be and respectively. Hence,

 𝑓(𝑥)=𝑥ℎ−1ℎ+𝑥ℎℎ.→→sinlimcoscoslimsin (1)

The other two limits are not as trivial. However, is actually a standard result. It is possible to derive this by proving that and applying the squeeze theorem. However, we will not give the details of that here. We can also note that for small , which also points to the fact that the limit will be equal to one. Notice also that, for this limit to evaluate to one, we require that be measured in radians. For this result, it is not the case if is measured in degrees.

We now turn our attention to the second limit,

We begin by multiplying the numerator and denominator by the conjugate of the numerator, which yields

Using the Pythagorean identity for sine and cosine, we can rewrite the numerator in terms of sine:

Applying the multiplicative law of limits, we have

Taking the limits, we have

This is not surprising given that, for small values of , can be approximated by

Substituting the value of these two limits into (1), we get

We will now consider the derivative of the cosine function. We could use a similar technique as we used for deriving the derivative of sine. However, rather than using the sum and difference trigonometric identities, we will use the product to sum formulae to demonstrate an alternative derivation.

### Example 2: Differentiating Cosine from First Principles

Given that , find from first principles.

Recall that the derivative of a function is defined by the limit

Substituting in cosine for , we have

To simplify the expression within the limit, we can use our knowledge of trigonometric identities. In particular, we can use the product to sum formula, to rewrite this expression as

Using the properties of limits, we can rewrite this as

The first of these limits simply evaluates to . As for the second, it is equivalent to . Using standard results, we find this is equal to one. Hence,

These formulae for the derivatives of sine and cosine can be generalized to the derivatives of and as follows:

To differentiate the other trigonometric functions, we will appeal to the rules of derivatives, in particular, to the quotient rule. Below is a summary of the product and quotient rules.

### Product Rule

Given two differentiable functions and , the derivative of their product is given by

This can be written succinctly using prime notation as follows:

### Quotient Rule

Given two differentiable functions and , the derivative of their quotient is given by

This can be written succinctly using prime notation as follows:

### Example 3: Derivative of the Tangent Function

Evaluate the rate of change of at .

Recall that we can find the rate of change of a given function at a given point by finding the value of its derivative at the point. Hence, we will first find the derivative of .

Using the trigonometric identity that , we can express as

To find the derivative of , we can use the quotient rule: setting and . We begin by differentiating and to find expressions for and . Starting with , we know the derivative of is . Hence,

Similarly,

Substituting these expressions into the quotient rule, we have

Using the Pythagorean identity for sine and cosine, we can simplify the numerator. Hence,

We can now substitute in to find the rate of change of at this point which gives

The previous example demonstrated that

Below is a graph of the tangent function and its derivative.

Thus far, we have looked at the derivatives of sine, cosine, and tangent. Below is a summary of their derivatives.

### Derivatives of Trigonometric Functions

The derivatives of the trigonometric functions are as follows:

Generally, you will be able to quote and use these results without derivation. However, you might be expected to derive these results using the techniques we have outlined above. Furthermore, it is certainly helpful to commit these to memory. This will dramatically increase your ability to tackle differentiation problems with ease and efficiency.

We will now consider a number of examples where we can apply these results without derivation.

### Example 4: Consecutive Derivatives of Sine

Find the thirty-third derivative of .

It would certainly be tedious to differentiate this function thirty-three times in a row. Therefore, we will begin by looking at the first few derivatives to see whether there is a pattern. The derivative of the sine function is the cosine function. Hence,

The cosine function differentiates to the negative sine function:

Continuing, we find which is equal to the function we started with. Hence, this pattern will repeat for higher-order derivatives. Therefore, the general rule is for . Since , we have

Using a similar argument where we set , we can derive the general formula where , for the higher-order derivatives of the cosine function.

### Example 5: Using the Product Rule with Trigonometric Functions

If , determine .

This function is a product of two differentiable functions and . We can, therefore, use the product rule which states that, for a function , to find its derivative. We begin by differentiating and . Using the rule for differentiating monomials, we have

Similarly, using the rule for differentiating sine, we have

We can now substitute these expressions into the formula for the product rule as follows:

### Example 6: Differentiating Trigonometric Functions

If , find .

When faced with a question like this, we might think about applying the product rule or the chain rule for composite functions. However, often times it is useful to think whether we can simplify the expression using trigonometric identities that we already know. This is the approach we will take to demonstrate how this significantly simplifies finding the derivative. We begin by factoring the 2 out of the parentheses:

We can now expand the parentheses as follows:

We can now use the Pythagorean identity for sine and cosine to simplify the expression as follows:

We can now use the double angle formula, to rewrite this as

We now have a much simpler expression for the function which we can easily differentiate using the rules of differentiation for trigonometric functions. Using the fact that and that the derivative of a constant term is zero, we have

The previous example demonstrates an important point regarding differentiation in general but is particularly poignant for trigonometric functions: we can often use the knowledge we have to manipulate the expression for the function to simplify the process of taking the derivative. We will finish by looking at one last example where we apply the quotient rule to evaluate the derivative.

### Example 7: Using the Quotient Rule with Trigonometric Functions

If , find .

The function we have been given is a quotient of two differential functions. Therefore, we can apply the quotient rule which states that, for a function , the derivative

Setting and , we begin by finding the derivatives of and . Using the rule of differentiation for trigonometric functions, we have and

Substituting these expressions back into the quotient rule, we have

Using the Pythagorean identity for sine and cosine, we can simplify the numerator as follows:

Finally, we cancel the common factor to get

### Key Points

1. We can extend our understanding of derivatives to trigonometric functions.
2. The derivatives of the sine, cosine, and tangent functions are as follows:
3. The higher-order derivatives of both the sine and the cosine functions form a repeating pattern. For sine, the pattern is as follows: For cosine, the pattern is
4. Using these standard results, we can find the derivatives of a large class of functions.