Ratios & Identities: Transformation Formula - Sum and Difference into Product

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables where both sides of the equation are defined. Transformation formulas are a set of trigonometric identities that allow us to express sums and differences of sines and cosines as products. These formulas are particularly useful in simplifying trigonometric expressions and solving trigonometric equations.

Transformation Formulas

The transformation formulas for converting sums and differences into products are as follows:

Sine Transformation Formulas

1. $\sin(A + B) = \sin A \cos B + \cos A \sin B$
2. $\sin(A - B) = \sin A \cos B - \cos A \sin B$

Using these, we can derive the product-to-sum identities:

• $\sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$
• $\sin A - \sin B = 2 \cos \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right)$

Cosine Transformation Formulas

1. $\cos(A + B) = \cos A \cos B - \sin A \sin B$
2. $\cos(A - B) = \cos A \cos B + \sin A \sin B$

Using these, we can derive the product-to-sum identities:

• $\cos A + \cos B = 2 \cos \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$
• $\cos A - \cos B = -2 \sin \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right)$

Table of Differences and Important Points

Identity Type	Sum Formulas	Difference Formulas	Product Formulas
Sine	$\sin(A + B) = \sin A \cos B + \cos A \sin B$	$\sin(A - B) = \sin A \cos B - \cos A \sin B$	$\sin A \pm \sin B = 2 \sin \left(\frac{A \pm B}{2}\right) \cos \left(\frac{A \mp B}{2}\right)$
Cosine	$\cos(A + B) = \cos A \cos B - \sin A \sin B$	$\cos(A - B) = \cos A \cos B + \sin A \sin B$	$\cos A \pm \cos B = 2 \cos \left(\frac{A \pm B}{2}\right) \cos \left(\frac{A \mp B}{2}\right)$

Examples

Let's look at some examples to understand how to apply these transformation formulas.

Example 1: Sine Sum into Product

Express $\sin 75^\circ + \sin 15^\circ$ as a product.

Using the formula $\sin A + \sin B = 2 \sin \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)$, we get:

$$ \sin 75^\circ + \sin 15^\circ = 2 \sin \left(\frac{75^\circ + 15^\circ}{2}\right) \cos \left(\frac{75^\circ - 15^\circ}{2}\right) \ = 2 \sin 45^\circ \cos 30^\circ \ = 2 \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) \ = \frac{\sqrt{6}}{2} $$

Example 2: Cosine Difference into Product

Express $\cos 70^\circ - \cos 20^\circ$ as a product.

Using the formula $\cos A - \cos B = -2 \sin \left(\frac{A + B}{2}\right) \sin \left(\frac{A - B}{2}\right)$, we get:

$$ \cos 70^\circ - \cos 20^\circ = -2 \sin \left(\frac{70^\circ + 20^\circ}{2}\right) \sin \left(\frac{70^\circ - 20^\circ}{2}\right) \ = -2 \sin 45^\circ \sin 25^\circ \ = -2 \left(\frac{\sqrt{2}}{2}\right) \sin 25^\circ \ = -\sqrt{2} \sin 25^\circ $$

Conclusion

Transformation formulas are powerful tools in trigonometry that allow us to convert between sums and differences of angles to products, which can simplify the process of solving trigonometric equations and proving identities. Understanding and applying these formulas is essential for students preparing for exams in mathematics, particularly in topics involving trigonometry.