Sum-to-Product Formulas
Convert sums into products.
Key idea
Convert sums into products.
In depth
sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2), and cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2). They turn sums into products, useful for solving equations and understanding beats.
Worked example
sin 75° + sin 15° = 2 sin 45° cos 30° = 2(√2/2)(√3/2) = √6/2.
Formula
Real-world uses
- Simplifying the superposition of waves and interference patterns in physics
Common mistake
Mixing up which of the average or half difference goes in which function.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Sum-to-Product Formulas?
Convert sums into products. It is a topic at the expert level in Trigonometry.
What is the formula for Sum-to-Product Formulas?
The key formula is sin a + sin b = 2 sin((a+b)/2) cos((a−b)/2).
Where is Sum-to-Product Formulas used in real life?
Real-world examples: Simplifying the superposition of waves and interference patterns in physics.
What should I learn before Sum-to-Product Formulas?
Make sure you are comfortable with Product-to-Sum Formulas, the previous topic in Trigonometry.
What is a common mistake with Sum-to-Product Formulas?
Mixing up which of the average or half difference goes in which function.
How can I practise Sum-to-Product Formulas?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Sum-to-Product Formulas?
Move on to Triple Angle Formulas next.
Related expert topics in Trigonometry
- Half Angle Formulas – Trig of θ/2.
- Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas – Convert products into sums.
- Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).
- Trig Equations with Identities – Use identities to factor trig equations.