# 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:

- 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:

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)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)## 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 $\sin x$.