## Circular Functions

Examples of Circular functions are sin(x), cos(x) and tan(x), they are also known as trigonometric functions.

### Differentiation of Sin

The derivative of sin is supplied in the formula sheet as:

$frac{d}{dx}(sin(ax))= acos(ax)$

When we find the derivative of a sin function we find the derivative of what is inside the brackets and multiply by cos of what was inside the sin brackets.

#### Example of Sin Differentiation

$frac{d}{dx}sin(2theta+5)$

Step 1 is to find the derivative of what is inside the brackets.

$frac{d}{dx}(2theta+5)newline =2$

Step 2  is to convert the sin function to a cos function while keeping what is in the brackets the same.

$sin(2theta+5) rightarrow cos(2theta+5)$

Step 3 is to multiply the first two steps together.

$2 times cos(2theta+5)newline = 2cos(2theta+5)$

### Differentiation of Cos

The derivative of sin is supplied in the formula sheet as:

$frac{d}{dx}(cos(ax))= -asin(ax)$

When we find the derivative of a cos function we find the derivative of what is inside the brackets and multiply by the negative sin of what was inside the cos brackets.

#### Example of Cos Differentiation

$frac{d}{dx}cos(4theta)$

Step 1 is to find the derivative of what is inside the brackets.

$frac{d}{dx}(4theta)newline =4$

Step 2  is to convert the sin function to a cos function while keeping what is in the brackets the same.

$cos(4theta) rightarrow -sin(4theta)$

Step 3 is to multiply the first two steps together.

$4times -sin(4theta)newline = -4sin(4theta)$

### Differentiation of Tan

The derivative of sin is supplied in the formula sheet as:

$frac{d}{dx}(tan(ax))= frac{a}{cos^{2}(ax)}=asec^{2}(ax)$

When we find the derivative of a sin function we find the derivative of what is inside the brackets and multiply by cos of what was inside the sin brackets.

#### Example of Tan Differentiation

$frac{d}{dx}tan(theta+7)$

Step 1 is to again find the derivative of what is inside the brackets.

$frac{d}{dx}(theta+7)newline =1$

Step 2  is to convert the tan function to a one over cos squared or a sec squared function while keeping what is in the brackets the same. In this instance we will just keep it in terms of cos, although either is correct.

$tan(theta+7) rightarrow frac{1}{cos^{2}(theta+7)}$

Step 3 is to multiply the first two steps together.

$1 times frac{1}{cos^{2}(theta+7)}newline = frac{1}{cos^{2}(theta+7)}$