Trigonometry Sheet 6
Transformation Formulas
1. Product to Sum Formulas
Transforming products of ratios into additions or subtractions.
| Product Form | Sum/Difference Form |
|---|---|
| $$2 \sin A \cos B$$ | $$\sin(A + B) + \sin(A – B)$$ |
| $$2 \cos A \sin B$$ | $$\sin(A + B) – \sin(A – B)$$ |
| $$2 \cos A \cos B$$ | $$\cos(A + B) + \cos(A – B)$$ |
| $$2 \sin A \sin B$$ | $$\cos(A – B) – \cos(A + B)$$ |
2. Sum to Product Formulas (C & D Formulas)
Useful for solving trigonometric equations and simplifying expressions.
| Sum/Difference Form | Product Form |
|---|---|
| $$\sin C + \sin D$$ | $$2 \sin \left( \frac{C+D}{2} \right) \cos \left( \frac{C-D}{2} \right)$$ |
| $$\sin C – \sin D$$ | $$2 \cos \left( \frac{C+D}{2} \right) \sin \left( \frac{C-D}{2} \right)$$ |
| $$\cos C + \cos D$$ | $$2 \cos \left( \frac{C+D}{2} \right) \cos \left( \frac{C-D}{2} \right)$$ |
| $$\cos C – \cos D$$ | $$2 \sin \left( \frac{C+D}{2} \right) \sin \left( \frac{D-C}{2} \right)$$ |