Section 4.4 Derivatives of trigonometric functions
In the previous section we proved a few foundational differentiation rules from the definition of the derivative. In this section we use the definition of the derivative and differentiation rules to calculate the derivatives of trigonometric functions. We encounter along the way a few interesting and non-trivial limits involving trigonometric functions.
Objectives
You should be able to:
- Determine the derivatives of the primary trigonometric functions, \(\sin x\) and \(\cos x\text{,}\) from the definition of derivatives, and recall the result.
- Calculate the derivatives of other trigonometric functions using the derivatives of the primary trigonometric functions, the quotient rule and trigonometric identities.
- Evaluate certain limits involving trigonometric functions, such as \(\displaystyle \lim_{x \to 0} \frac{\sin x}{x}\) and \(\displaystyle \lim_{x \to 0} \frac{\cos x - 1}{x}\text{.}\)
Subsection 4.4.1 Instructional video
Subsection 4.4.2 Key concepts
Concept 4.4.1. Useful trigonometric limits.
\begin{align*}
\lim_{x \to 0} \frac{\sin x}{x} =\amp 1\\
\lim_{x \to 0} \frac{\cos x - 1}{x} =\amp 0.
\end{align*}
The first one can be proved using the Squeeze Theorem; the second one then follows using trigonometric identities and limit laws.
Concept 4.4.2. Derivatives of trigonometric functions.
\begin{align*}
\frac{d}{dx} \sin x =\amp \cos x,\amp \frac{d}{dx} \cos x =\amp - \sin x,\\
\frac{d}{dx} \tan x =\amp \sec^2 x,\amp \frac{d}{dx} \cot x =\amp - \csc^2 x,\\
\frac{d}{dx} \sec x =\amp \sec x \tan x,\amp \frac{d}{dx} \csc x =\amp - \csc x \cot x.
\end{align*}
The first two can be proved from the definition of the derivative; the other ones then follow using the quotient rule.
Further readings 4.4.3 Further readings
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