Product-to-Sum and Sum-to-Product Formulas in Trigonometry

Product-to-Sum and Sum-to-Product Formulas

Introduction

Product-to-sum and sum-to-product formulas are essential tools in trigonometry, allowing the transformation of products of trigonometric functions into sums or differences, and vice versa. These identities simplify complex trigonometric expressions and are widely used in integration, solving equations, and proving other identities.

Product-to-Sum Formulas

Definition and Formulas

The product-to-sum formulas express products of sines and cosines as sums or differences of trigonometric functions. These are derived from the sum and difference identities for sine and cosine.

• sin α sin β: $\sin \alpha \sin \beta = \frac{1}{2} \left[ \cos(\alpha - \beta) - \cos(\alpha + \beta) \right]$

• cos α cos β: $\cos \alpha \cos \beta = \frac{1}{2} \left[ \cos(\alpha - \beta) + \cos(\alpha + \beta) \right]$

• sin α cos β: $\sin \alpha \cos \beta = \frac{1}{2} \left[ \sin(\alpha + \beta) + \sin(\alpha - \beta) \right]$

Example: Applying Product-to-Sum

• Example: Express $\cos(4\theta) \cos(6\theta)$ as a sum.

  • Using the formula: $\cos \alpha \cos \beta = \frac{1}{2} [\cos(\alpha - \beta) + \cos(\alpha + \beta)]$

  • Here, $\alpha = 4\theta$, $\beta = 6\theta$

  • So, $\cos(4\theta) \cos(6\theta) = \frac{1}{2} [\cos(-2\theta) + \cos(10\theta)] = \frac{1}{2} [\cos(2\theta) + \cos(10\theta)]$

Sum-to-Product Formulas

Definition and Formulas

The sum-to-product formulas allow us to write sums or differences of sines and cosines as products. These are especially useful for simplifying trigonometric expressions and solving equations.

• sin α + sin β: $\sin \alpha + \sin \beta = 2 \sin \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right)$

• sin α − sin β: $\sin \alpha - \sin \beta = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\alpha - \beta}{2} \right)$

• cos α + cos β: $\cos \alpha + \cos \beta = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right)$

• cos α − cos β: $\cos \alpha - \cos \beta = -2 \sin \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\alpha - \beta}{2} \right)$

Example: Applying Sum-to-Product

• Example: Express $\sin(6\theta) - \sin(4\theta)$ as a product.

  • Using the formula: $\sin \alpha - \sin \beta = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\alpha - \beta}{2} \right)$

  • Here, $\alpha = 6\theta$, $\beta = 4\theta$

  • So, $\sin(6\theta) - \sin(4\theta) = 2 \cos(5\theta) \sin(\theta)$

Worked Examples

Finding Exact Values Using Product-to-Sum

• Example: Find the exact value of $\sin 165^\circ \cdot \sin 45^\circ$.

  • Using the formula: $\sin \alpha \sin \beta = \frac{1}{2} [\cos(\alpha - \beta) - \cos(\alpha + \beta)]$

  • Substitute $\alpha = 165^\circ$, $\beta = 45^\circ$:

  • $\sin 165^\circ \sin 45^\circ = \frac{1}{2} [\cos(120^\circ) - \cos(210^\circ)]$

  • Evaluate: $\cos(120^\circ) = -\frac{1}{2}$, $\cos(210^\circ) = -\frac{\sqrt{3}}{2}$

  • So, $\sin 165^\circ \sin 45^\circ = \frac{1}{2} \left[ -\frac{1}{2} + \frac{\sqrt{3}}{2} \right] = \frac{1}{2} \cdot \frac{-1 + \sqrt{3}}{2} = \frac{-1 + \sqrt{3}}{4}$

Expressing Products as Sums

• Example: Express $\cos(4\theta) \cos(6\theta)$ as a sum.

  • As shown above, $\cos(4\theta) \cos(6\theta) = \frac{1}{2} [\cos(2\theta) + \cos(10\theta)]$

Expressing Sums or Differences as Products

• Example: Express $\sin(6\theta) - \sin(4\theta)$ as a product.

  • As shown above, $\sin(6\theta) - \sin(4\theta) = 2 \cos(5\theta) \sin(\theta)$

Establishing Trigonometric Identities

Example: Proving an Identity

• Example: Show that $\cos \theta + \cos 3\theta = 2 \cos 2\theta \cos \theta$

  • Using the sum-to-product formula: $\cos \alpha + \cos \beta = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right)$

  • Here, $\alpha = \theta$, $\beta = 3\theta$

  • $\cos \theta + \cos 3\theta = 2 \cos(2\theta) \cos(-\theta) = 2 \cos(2\theta) \cos(\theta)$

Summary Table: Product-to-Sum and Sum-to-Product Formulas

Product-to-Sum	Sum-to-Product
$\sin \alpha \sin \beta = \frac{1}{2} [\cos(\alpha - \beta) - \cos(\alpha + \beta)]$	$\sin \alpha + \sin \beta = 2 \sin \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right)$
$\cos \alpha \cos \beta = \frac{1}{2} [\cos(\alpha - \beta) + \cos(\alpha + \beta)]$	$\sin \alpha - \sin \beta = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\alpha - \beta}{2} \right)$
$\sin \alpha \cos \beta = \frac{1}{2} [\sin(\alpha + \beta) + \sin(\alpha - \beta)]$	$\cos \alpha + \cos \beta = 2 \cos \left( \frac{\alpha + \beta}{2} \right) \cos \left( \frac{\alpha - \beta}{2} \right)$
	$\cos \alpha - \cos \beta = -2 \sin \left( \frac{\alpha + \beta}{2} \right) \sin \left( \frac{\alpha - \beta}{2} \right)$

Additional info: These formulas are especially useful in calculus for integrating products of trigonometric functions and in simplifying trigonometric equations for solution.