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

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.