In this Article we will understand the formula of a sine function **sin(3x), **the formula of sin3x is as follows:

**Sin(3x) = 3 sinx – 4sin^{3}x**

Now we will understand every concept of trigonomatry related to this function one by one, like how it is derived, what are its properties etc.

## What is a Trigonometric function:

Trigonometry explains the relation between angles and the corresponding sides of triangles

Trigonometric functions, on the other hand defines these relationships as ratios.

## Trigonometric Function Sin(x):

The Sine function, written as sin(x), is a trigonometric function that maps an angle to the ratio of the length of the side opposite the angle to the hypotenuse of a right triangle. The value of sine function oscillates between (-1,1).

## Some Trigonometric identities:

**Double Angle Formula for Sine:***sin(2x) = 2sin(x)cos(x)***Sum-to-Product Formula for Sine:**sin( A + B ) = sin( A ) cos(B) + cos(A)sin(B)**Double Angle Formula for cosine:***cos( 2x) = 1- 2sin*^{2}(x)

## Derivation of sin(3x) Formula:

Step 1: we need to rewrite the sin(3x) as:

*sin( 3x ) = sin ( 2x + x )*

Step 2: Here we use the Sum-to-product Formula of sine as:

*sin( 3x ) = sin( 2x )cos( x ) + cos( 2x )sin( x )*

Step 3: we already know the double-angle Formula of sine and cosine, so we replace the sin(2x) and cos(2x) with their respective values as:

*sin ( 3x ) = {2sin( x )cos( x )}cos( x ) + {1 – 2sin ^{2}( x )}sin( x )*

Step 4: Simplify the equation by multiplying the terms

*sin ( 3x ) = 2sin( x )cos ^{2}( x ) + sin( x ) – 2sin^{3}( x )*

Step 5: Use the value of cos^{2}(x) as (1-sin^{2}(x))

*sin( 3x ) = 2sin( x ){1 – sin ^{2}( x )} + sin( x ) – 2sin^{3}( x )*

Step 6: Simplify and combine the like terms:

## Example:

Q: Find the value of **sin(3π/4)**

To find **sin(3π/4)**, we can use the derived formula for **sin(3x):**

*sin(3x) = 3 sin(x) – 4sin3(x)*

In this case, **x is π/4**. So, we can substitute π/4 into the formula:

*sin(3π/4) = 3 sin(π/4) – 4sin3(π/4)*

Now we know that: *sin(π/4) = √ (2)/2*

Substitute the value of sin(π/4) in the above equation we get:

*sin(3π/4) = √ (2)*

## Derivative of Sin(3x):

To find the derivative of sin(3x) we will first use the direct method as:

*f(x) = sin(3x)*

*f'(x) = d /dx (sin(3x))*

*f'(x) = cos(3x)*d/dx(3x)*

*f'(x) = cos(3x)*3*

*f'(x) = 3cos(3x)*

## Derivative of sin(3x) using first principle:

The expression for calculating derivative using first principle is as

*f'(x) = limt { f(x+h) – f(x) }/ h*