Derivatives of Trigonometric Functions

# Derivatives of Sine, Cosine, and Tangent

Some common functions that appear in equations are the basic trigonometric functions. The following three theorems will establish their derivatives.

 Theorem 1: If $f(x) = \sin x$ then $\frac{d}{dx} f(x) = \cos x$.
• Proof: Let $f(x) = \sin x$. From the definition of a derivative it follows that:
(1)
\begin{align} f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \\ f'(x) = \lim_{h \to 0} \frac{\sin(x + h) - \sin(x)}{h} \\ f'(x) = \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h - \sin x}{h} \\ f'(x) = \lim_{h \to 0} \left ( \sin x \frac{(\cos h - 1)}{h} + \cos x \frac{\sin h}{h} \right ) \end{align}
• Now recall that $\lim_{h \to 0} \frac{(\cos h - 1)}{h} = 0$, and $\lim_{h \to 0} \frac{\sin h}{h} = 1$, so therefore, $f'(x) = 0 \cdot \sin x + 1 \cdot \cos x = \cos x$. $\blacksquare$
 Theorem 2: If $f(x) = \cos x$ then $\frac{d}{dx} f(x) = -\sin x$.
• Proof of Property (b): Let $f(x) = \cos x$. From the definition of a derivative it follows that:
(2)
\begin{align} f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \\ f'(x) = \lim_{h \to 0} \frac{\cos(x + h) - \cos x}{h} \\ f'(x) = \lim_{h \to 0} \frac{\cos x \cos h - \sin x \sin h - \cos x}{h} \\ f'(x) = \lim_{h \to 0} \cos x \frac{(\cos h - 1)}{h} - \lim_{h \to 0} \sin x \frac{\sin h}{h} \\ f'(x) = 0 \cdot \cos x - 1 \cdot \sin x \\ f'(x) = -\sin x \quad \blacksquare \end{align}
 Theorem 3: If $f(x) = \tan x$ then $\frac{d}{dx} f(x) = \sec ^2 x$.

We will now prove theorem 3 at this point and time.

## Example 1

Differentiate $f(x) = 3 \cos x + 2 \sin x$.

Applying the theorems above and we get that $f'(x) = -3 \sin x + 2 \cos x$.

## Example 2

Differentiate $f(x) = 2\tan x - 2 \sin x$.

Applying the theorems above and we get that $f'(x) = 2 \sec ^2 x - 2 \cos x$.

# Cyclic Pattern of Higher Order Derivatives of Sine and Cosine

Notice that the derivative of $\sin x$ is $\cos x$… the derivative of $\cos x$ is $-\sin x$, etc…. Hopefully you can see that there is a cycle here as:

(3)
\begin{align} \frac{d}{dx} \sin x = \cos x \end{align}
(4)
\begin{align} \frac{d}{dx} \cos x = -\sin x \end{align}
(5)
\begin{align} \frac{d}{dx} -\sin x = -\cos x \end{align}
(6)
\begin{align} \frac{d}{dx} -\cos x = \sin x \end{align}

We can thus make a general rule for higher order derivatives of $\sin x$ and $\cos x$. If $f(x) = \sin x$ or $f(x) = \cos x$ and $n = 4k + 1$, then for all integers $k ≥ 0$:

(7)
\begin{align} \frac{d^n}{dx^n} f(x) = f(x) \end{align}

## Example 3

What function do you get if you differentiate $f(x) = \sin x$ one-hundred times?

Notice that every four differentiations, we will get back to $\sin x$. Therefore, if we differentiate $100$ times, we will still have $\sin x$.