Product-to-Sum Formulas
Convert products into sums.
Key idea
Convert products into sums.
In depth
sin A cos B = ½[sin(A + B) + sin(A − B)], cos A cos B = ½[cos(A − B) + cos(A + B)], sin A sin B = ½[cos(A − B) − cos(A + B)]. They turn products into sums, easier to integrate.
Worked example
sin 3x cos x = ½(sin 4x + sin 2x).
Formula
Real-world uses
- Analysing beat patterns and mixing two radio signals in communication engineering
Common mistake
Forgetting the ½ factor.
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 Product-to-Sum Formulas?
Convert products into sums. It is a topic at the expert level in Trigonometry.
What is the formula for Product-to-Sum Formulas?
The key formula is sin a cos b = ½[sin(a+b)+sin(a−b)].
Where is Product-to-Sum Formulas used in real life?
Real-world examples: Analysing beat patterns and mixing two radio signals in communication engineering.
What should I learn before Product-to-Sum Formulas?
Make sure you are comfortable with Power Reducing Formulas, the previous topic in Trigonometry.
What is a common mistake with Product-to-Sum Formulas?
Forgetting the ½ factor.
How can I practise Product-to-Sum 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 Product-to-Sum Formulas?
Move on to Sum-to-Product 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.
- Sum-to-Product Formulas – Convert sums into products.
- 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.