# 2 Sin A Cos A Formula — Sin 2A Proof, Examples  
### TL;DR  
The 2 sin A cos A formula states that 2sin⁡Acos⁡A=sin⁡2A, the double-angle identity for sine. This article gives the formula, its proof from the sine sum formula, its tangent form, six worked examples in degrees and radians, the unit-circle picture, the most common sign mistakes, and FAQs.
  
## The Identity That Folds Two Factors Into One Angle  
A radar dish sweeping a beam, a pendulum at the bottom of its swing, an AC generator at peak output — each is governed by a quantity that doubles an angle, and the **2 sin A cos A formula** is what collapses that doubling into a single clean term.

### The **2 sin A cos A formula** says:  
\[ 2\sin A \cos A = \sin 2A \]  
In words: twice the product of the sine and cosine of an angle equals the sine of double that angle. It is one of the three double-angle identities (the sine one), and it falls directly out of the sum and difference formulas when the two angles are equal.

## What Does 2 Sin A Cos A Equal?  
The expression 2sin⁡Acos⁡A equals sin⁡2A — the sine of twice the angle. This holds for **every** real value of A, in degrees or radians, with no domain restriction. The factor of 2 is crucial; it ensures the two sides match exactly rather than off by half.

## How Is the 2 Sin A Cos A Formula Proved?  
The proof is one substitution into a formula you already have.  
**From the sine sum formula.** The sine of a sum is:  
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B. \]  
Set B=A, then:  
\[ \sin(A + A) = \sin 2A \]  
This leads to:  
\[ \sin A \cos A + \cos A \sin A = 2\sin A \cos A. \]  
Equating the two gives:  
\[ \sin 2A = 2\sin A \cos A. \]  
That is the whole derivation — the double-angle identity is the angle-sum identity looking at itself.

## Double-Anchoring — Right Triangle and Unit Circle  
**From the right triangle.** Take an acute angle A in a right triangle with hypotenuse 1. Then sin⁡A is the opposite side and cos⁡A is the adjacent side. The product 2sin⁡Acos⁡A is twice the area of that triangle, which equals the height sin⁡2A on the unit circle.

**From the unit circle.** Place angle A so its terminal point is (cos⁡A, sin⁡A). The angle 2A has terminal point (cos⁡2A, sin⁡2A). The y-coordinate of that doubled angle — its sine — is exactly 2sin⁡Acos⁡A.

## The Tangent Form  
The identity can also be rewritten using tangent:  
\[ \sin 2A = 2\sin A \cos A = \frac{2\tan A}{1 + \tan^2 A}. \]  
This form is useful in calculus substitutions.

## Examples of the 2 Sin A Cos A Formula  
### Example 1  
**Simplify 2sin⁡30°cos⁡30°.**  
By the formula, this is sin⁡(2×30°)=sin⁡60°=\[ \frac{\sqrt{3}}{2}. \]  
  
### Example 2  
**Evaluate 2sin⁡15°cos⁡15°.**  
Using the identity:  
\[ 2\sin 15° \cos 15° = \sin(2×15°) = \sin 30° = \frac{1}{2}. \]  
  
### Example 3  
**Find sin⁡120° using the formula.**  
Write 120°=2×60°, so sin⁡120°=2sin⁡60°cos⁡60°. Substitute:  
\[ \sin 120° = 2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2}. \]  
  
### Example 4  
**Given sin⁡A=\frac{3}{5} and cos⁡A=\frac{4}{5}, find sin⁡2A.**  
Using the formula:  
\[ \sin 2A = 2\sin A \cos A = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25}. \]  
  
### Example 5  
**Use the tangent form to find sin⁡2A when tan⁡A=\frac{1}{2}.**  
\[ \sin 2A = \frac{2 \cdot \frac{1}{2}}{1 + \frac{1}{4}} = \frac{4}{5}. \]  
  
### Example 6  
**Integrate ∫2sin⁡xcos⁡x,dx.**  
Recognise the integrand:  
\[ \int 2\sin x \cos x \, dx = \int \sin 2x \, dx = -\frac{1}{2}\cos 2x + C. \]

## Where the Sine Double-Angle Identity Earns Its Keep  
This identity is not a textbook curiosity — it is crucial in the real world applications, like projectile range, AC power calculations, and signal processing.

## The Mathematicians Behind the 2 Sin A Cos A Formula  
**Claudius Ptolemy** and **Bhaskara II** were instrumental in deriving and formalising concepts related to this identity.

## Why Students Get the Sine Double Angle Wrong  
### Mistake 1  
They often drop the cosine factor, treating doubling the angle as doubling the sine.  
### Mistake 2  
Confusing sin⁡2A with sin²A, leading to misunderstandings in calculations.  
### Mistake 3  
Mixing degree and radian values mid-problem, resulting in incorrect values.

## Key Takeaways  
- The **2 sin A cos A formula** is crucial in many mathematical and physical applications.  
- The most common mistake is incorrectly dropping the cosine factor, which can lead to impossible values.