# Sin A + Sin B Formula — Proof and Examples

## TL;DR  
The sin A + sin B formula is the sum-to-product identity:  
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]  
It converts the sum of two sines into the product of a sine and a cosine. The proof uses the angle-sum identities  
\[ \sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta \]  
— adding two of them and substituting \( \alpha + \beta = A \) and \( \alpha - \beta = B \) collapses the algebra into the product form.

## When Two Notes Become One Beat  
Strike two adjacent piano keys — A and B-flat — and you hear something neither key produces alone: a slow throbbing called a "beat." That beat is  
\[ \sin A + \sin B \]  
converted by ear into the product form  
\[ 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]  
Piano tuners use this exact identity — without ever writing it down — to tune by ear.

## What Is the Sin A + Sin B Formula?  
The **sin A + sin B formula** is one of four sum-to-product identities in trigonometry. It states:  
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]

The four sum-to-product identities together are the inverse of the four product-to-sum identities — they let you convert between additive and multiplicative forms of trig expressions, solving integration problems, simplifying wave-physics calculations, and powering band-pass filters in every radio.

## Proof of the Sin A + Sin B Formula  
Start with the angle-sum and angle-difference identities for sine:  
\[ \sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta \tag{1} \]  
\[ \sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta \tag{2} \]

**Add equation (1) to equation (2)**:  
\[ \sin(\alpha + \beta) + \sin(\alpha - \beta) = 2 \sin\alpha\cos\beta \]

Now substitute new variables: let \( \alpha + \beta = A \) and \( \alpha - \beta = B \):  
\[ \alpha = \frac{A+B}{2}, \quad \beta = \frac{A-B}{2} \]  
Substituting back:  
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]

## The Four Sum-to-Product Identities  
The sin A + sin B formula doesn't live alone. There are four sum-to-product identities:  
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]  
\[ \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]  
\[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]  
\[ \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]

## Sum-to-Product vs Product-to-Sum — Comparison Table  
| Direction   | What you start with           | What you get                          | Use when                                     |
|-------------|-------------------------------|---------------------------------------|----------------------------------------------|
| **Sum-to-product** | \( \sin A + \sin B \)           | \( 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \) | Solving equations, finding common factors, beat-frequency physics |
| **Product-to-sum**    | \( \sin A \cdot \sin B \) | \( \frac{1}{2}[\cos(A-B) - \cos(A+B)] \)   | Integration, Fourier analysis, signal modulation |

## Three Worked Examples — Quick, Standard, Stretch  
### Quick  
**Express \( 
\sin 75° + \sin 15° \) as a single product.**  
Using the formula with \( A=75° \), \( B=15° \):  
\[ A+B=90°, A-B=30° \]  
\[ \sin 75° + \sin 15° = 2 \sin 45° \cos 30° = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2} \]

### The Detour Students Take — Standard Example  
**Simplify \( 
\sin 4x + \sin 2x\) .**  
The correct path: Apply the sin A + sin B formula with \( A=4x \), \( B=2x \):  
\[ \sin 4x + \sin 2x = 2 \sin 3x \cos x \]

### Stretch  
**A beat is heard when two tuning forks at 440 Hz and 442 Hz vibrate simultaneously. What is the beat frequency?**  
Apply the sum-to-product formula with A and B as the frequencies:  
\[ A+B=441, A-B=1\]  
The perceived tone is 441 Hz, and the beat frequency is 2 Hz.

## Key Takeaways  
- The **sin A + sin B formula** is  \( 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \).  
- One of four sum-to-product identities; their inverses are the four product-to-sum identities.

## Sharpen Your Sin A + Sin B Skills — Three Practice Problems  
1. Express \( \cos 75° + \cos 15°\) as a single product, then evaluate exactly.  
2. Simplify \( \sin 6x - \sin 4x\) to a product form.  
3. Show that \( \sin(A+B) + \sin(A-B) = 2 \sin A \cos B\).

**If any didn’t give you the correct answer, re-check the corresponding formula.**