The Formula for sin(2θ)

Introduction:
The sine function, denoted as sin(θ), is a trigonometric function that relates the angles of a right triangle to the ratio of the length of its sides. It is widely used in various mathematical and scientific fields. The double-angle formula for sine, sin(2θ), allows us to find the value of sine for double the given angle.

Formula:
The formula for sin(2θ) is derived from the sum or difference formula for sine, which states that sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B). By substituting A = B = θ, we can obtain the formula for sin(2θ).

The double-angle formula for sine is given by:

sin(2θ) = 2sin(θ)cos(θ)

Explanation:
To understand the derivation of the formula, we'll break it down into smaller steps.

Step 1: Express sin(2θ) using the sum formula:
We start with the sum formula for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). By substituting A = B = θ, the formula becomes:

sin(2θ) = sin(θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ)

Step 2: Simplify the expression:
Since sin(θ)cos(θ) is the same as cos(θ)sin(θ), we can rewrite the formula as:

sin(2θ) = 2sin(θ)cos(θ)

This is the final formula for sin(2θ).

Example:
Let's consider an example to illustrate the usage of the formula. Suppose we want to find the value of sin(2π/3).

Using the formula sin(2θ) = 2sin(θ)cos(θ), we can substitute θ = π/3:

sin(2π/3) = 2sin(π/3)cos(π/3)

The values of sin(π/3) and cos(π/3) can be determined using the unit circle or trigonometric tables. In this case, sin(π/3) = √3/2 and cos(π/3) = 1/2. Plugging these values into the formula:

sin(2π/3) = 2 * (√3/2) * (1/2) = √3/2

Therefore, sin(2π/3) is equal to √3/2.

Conclusion:
The double-angle formula for sine, sin(2θ) = 2sin(θ)cos(θ), allows us to find the value of sine for double the given angle. It is derived from the sum or difference formula for sine and is widely used in trigonometry, physics, and engineering to solve problems involving angles and triangles.