Trigonometric Differentiation

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.

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

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