Products, Sums & Beats

Turning products into sums and back

An interactive lesson with 6 steps, about 5 minutes: you predict, drag and solve, with instant feedback on every answer. It runs in your browser and needs JavaScript.

What you’ll learn

• $2\cos A\cos B = \cos(A - B) + \cos(A + B)$; $\;2\sin A\sin B = \cos(A - B) - \cos(A + B)$.
• $\sin P + \sin Q = 2\sin\frac{P + Q}{2}\cos\frac{P - Q}{2}$.
• Two close frequencies beat at their difference frequency.

Steps

1. Product to sum
2. An exact product
3. Sum to product
4. Make the beats
5. Tuning a guitar
6. A sum of sines