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Section5.12The Derivatives of Trigonometric Functions

Subsection5.12.1Essential Trigonometric Identities

When we found the derivative of elementary exponential functions, we found that we needed to use a rule to rewrite \(b^{x+h}=b^x \cdot b^h\). This type of rule is called an identity. Identities provide rules to rewrite a formula in another form without changing the value of the formula. Trigonometric functions are all defined in terms of the elementary sine and cosine functions. Consequently, we need the basic identities of sine and cosine.

We start with the sum identities.

Proof

Next, we state the symmetries of sine and cosine.

Proof

Finally, because the sine and cosine are defined on a unit circle (with radius 1), we have a Pythagorean identity regarding the sum of the squared values.

Proof

Subsection5.12.2Differentiation of Sine and Cosine

We start by computing the derivatives for sine and cosine at the origin, \(\sin'(0)\) and \(\cos'(0)\). Once we know those values, we will be able to find the derivatives as functions.

Proof

It is important to note that getting the value for this limit to be the value 1 was a consequence of measuring angles in radians. For any other way that we might measure angles, the fraction of the circles area would be a ratio of the value \(x\) to the measurement of the angle to complete a full circle. For example, if we measured angles in degrees, we would have instead found \(\sin_{\mathrm{deg}}'(0) = \frac{2\pi}{360}\). Mathematically, one justification for measuring angles in radians is simply in order to guarantee that this \(\sin'(0)=1\).

Proof

Knowing the instantaneous rates of change of sine and cosine at \(x=0\) allows us to compute the derivative at any input value. The proofs for these differentiation rules rely on the sum identities for trigonometric functions.

Proof
Proof

Subsection5.12.3Derivatives of Other Trigonometric Functions

All other trigonometric functions are defined in terms of the sine and cosine functions. Knowing the derivatives of sine and cosine allow us to compute the derivative rules for each of the other trigonometric functions.

Proof

Subsection5.12.4Practice with Derivatives

When we take into account the chain rule, we have the following general derivative rules for trigonometric functions. Notice that cosine, cotangent, and cosecant all have a negative sign. Also note the similarity in formulas between the derivatives for sine and cosine, for tangent and cotangent, and for secant and cosecant. There are really only three differentiation rules, each with a complementary rule for the complementary functions.

General Derivative Rules for Trigonometric Functions

Let \(u\) represent any expression that depends on \(x\).\begin{align*} \frac{d}{dx}[\sin(u)] &= \cos(u) \frac{du}{dx} \\ \frac{d}{dx}[\cos(u)] &= -\sin(u) \frac{du}{dx} \\ \frac{d}{dx}[\tan(u)] &= \sec^2(u) \frac{du}{dx} \\ \frac{d}{dx}[\cot(u)] &= -\csc^2(u) \frac{du}{dx} \\ \frac{d}{dx}[\sec(u)] &= \sec(u) \tan(u) \frac{du}{dx} \\ \frac{d}{dx}[\csc(u)] &= -\csc(u) \cot(u) \frac{du}{dx} \end{align*}

The following examples illustrate how these rules can be used with other rules of differentiation.

Example5.12.9

Find \(\displaystyle \frac{d}{dx}[3\sin(x^2)]\).

Solution
Example5.12.10

Find \(\displaystyle \frac{d}{dx}[\sec(e^{3x})]\).

Solution
Example5.12.11

Find \(\displaystyle \frac{d}{dx}[e^{-3x}\sin(5x)]\).

Solution

Subsection5.12.5The Squeeze Theorem for Limits