Product-to-Sum and Sum-to-Product Formulas in Trigonometry 5.4

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

Introduction

Product-to-sum and sum-to-product formulas are important trigonometric identities that allow us to rewrite products of sines and cosines as sums or differences, and vice versa. These formulas are especially useful for simplifying trigonometric expressions and solving trigonometric equations.

Product-to-Sum Formulas

Definition and Formulas

• Product-to-sum formulas express products of sine and cosine functions as sums or differences of trigonometric functions.

• These are derived from the sum and difference identities for sine and cosine.

The main product-to-sum formulas are:

• $\sin x \sin y = \frac{1}{2} [ \cos(x-y) - \cos(x+y) ]$

• $\cos x \cos y = \frac{1}{2} [ \cos(x-y) + \cos(x+y) ]$

• $\sin x \cos y = \frac{1}{2} [ \sin(x+y) + \sin(x-y) ]$

• $\cos x \sin y = \frac{1}{2} [ \sin(x+y) - \sin(x-y) ]$

Examples

• Example 1: Simplify $\sin(\theta/6) \sin(\theta/4)$ using a product-to-sum formula.

  • Let $x = \theta/6$, $y = \theta/4$.

  • Apply the formula: $\sin x \sin y = \frac{1}{2} [ \cos(x-y) - \cos(x+y) ]$

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

  • Simplify the angles: $\theta/4 - \theta/6 = (3\theta - 2\theta)/12 = \theta/12$ and $\theta/4 + \theta/6 = (3\theta + 2\theta)/12 = 5\theta/12$

  • Final answer: $\frac{1}{2} [ \cos(\theta/12) - \cos(5\theta/12) ]$

• Example 2: Simplify $\cos(\theta) \cos(3\theta)$.

  • Let $x = 3\theta$, $y = \theta$.

  • Apply the formula: $\cos x \cos y = \frac{1}{2} [ \cos(x-y) + \cos(x+y) ]$

  • $= \frac{1}{2} [ \cos(3\theta - \theta) + \cos(3\theta + \theta) ]$

  • $= \frac{1}{2} [ \cos(2\theta) + \cos(4\theta) ]$

• Example 3: Simplify $\cos(3\theta) \sin(5\theta)$.

  • Rewrite as $\sin(5\theta) \cos(3\theta)$ for positive angles.

  • Apply the formula: $\sin x \cos y = \frac{1}{2} [ \sin(x+y) + \sin(x-y) ]$

  • $= \frac{1}{2} [ \sin(5\theta + 3\theta) + \sin(5\theta - 3\theta) ]$

  • $= \frac{1}{2} [ \sin(8\theta) + \sin(2\theta) ]$

• Example 4: Find the exact value of $\sin(19\pi/12) \cos(\pi/12)$.

  • Apply the formula: $\sin x \cos y = \frac{1}{2} [ \sin(x+y) + \sin(x-y) ]$

  • $= \frac{1}{2} [ \sin(19\pi/12 + \pi/12) + \sin(19\pi/12 - \pi/12) ]$

  • $= \frac{1}{2} [ \sin(20\pi/12) + \sin(18\pi/12) ]$

  • Simplify: $20\pi/12 = 5\pi/3$, $18\pi/12 = 3\pi/2$

  • Use unit circle values: $\sin(5\pi/3) = -\sqrt{3}/2$, $\sin(3\pi/2) = -1$

  • Final answer: $\frac{1}{2} [ -\sqrt{3}/2 + (-1) ] = \frac{1}{2} ( -\sqrt{3}/2 - 1 ) = -\frac{\sqrt{3}}{4} - \frac{1}{2}$

Sum-to-Product Formulas

Definition and Formulas

• Sum-to-product formulas express sums or differences of sines and cosines as products of trigonometric functions.

• These are useful for simplifying expressions and solving equations involving sums or differences of trigonometric functions.

The main sum-to-product formulas are:

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

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

• $\cos x + \cos y = 2 \cos\left( \frac{x+y}{2} \right) \cos\left( \frac{x-y}{2} \right)$

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

Examples

• Example 1: Simplify $\sin(5\theta) - \sin(3\theta)$.

  • Apply the formula: $\sin x - \sin y = 2 \cos\left( \frac{x+y}{2} \right) \sin\left( \frac{x-y}{2} \right)$

  • $= 2 \cos\left( \frac{5\theta + 3\theta}{2} \right) \sin\left( \frac{5\theta - 3\theta}{2} \right)$

  • $= 2 \cos(4\theta) \sin(\theta)$

• Example 2: Simplify $\cos(3\theta) + \cos(2\theta)$.

  • Apply the formula: $\cos x + \cos y = 2 \cos\left( \frac{x+y}{2} \right) \cos\left( \frac{x-y}{2} \right)$

  • $= 2 \cos\left( \frac{3\theta + 2\theta}{2} \right) \cos\left( \frac{3\theta - 2\theta}{2} \right)$

  • $= 2 \cos(2.5\theta) \cos(0.5\theta)$

• Example 3: Simplify $\sin(\theta) + \sin(5\theta)$.

  • Apply the formula: $\sin x + \sin y = 2 \sin\left( \frac{x+y}{2} \right) \cos\left( \frac{x-y}{2} \right)$

  • $= 2 \sin(3\theta) \cos(2\theta)$

• Example 4: Simplify $\cos(7\theta) - \cos(4\theta)$.

  • Apply the formula: $\cos x - \cos y = -2 \sin\left( \frac{x+y}{2} \right) \sin\left( \frac{x-y}{2} \right)$

  • $= -2 \sin(5.5\theta) \sin(1.5\theta)$

• Example 5: Find the exact value of $\sin 75^\circ + \sin 15^\circ$.

  • Apply the formula: $\sin x + \sin y = 2 \sin\left( \frac{x+y}{2} \right) \cos\left( \frac{x-y}{2} \right)$

  • $= 2 \sin(45^\circ) \cos(30^\circ)$

  • Use known values: $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, $\cos(30^\circ) = \frac{\sqrt{3}}{2}$

  • Final answer: $2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}$

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

Type	Formula
Product-to-Sum	$\sin x \sin y = \frac{1}{2} [ \cos(x-y) - \cos(x+y) ]$
Product-to-Sum	$\cos x \cos y = \frac{1}{2} [ \cos(x-y) + \cos(x+y) ]$
Product-to-Sum	$\sin x \cos y = \frac{1}{2} [ \sin(x+y) + \sin(x-y) ]$
Product-to-Sum	$\cos x \sin y = \frac{1}{2} [ \sin(x+y) - \sin(x-y) ]$
Sum-to-Product	$\sin x + \sin y = 2 \sin\left( \frac{x+y}{2} \right) \cos\left( \frac{x-y}{2} \right)$
Sum-to-Product	$\sin x - \sin y = 2 \cos\left( \frac{x+y}{2} \right) \sin\left( \frac{x-y}{2} \right)$
Sum-to-Product	$\cos x + \cos y = 2 \cos\left( \frac{x+y}{2} \right) \cos\left( \frac{x-y}{2} \right)$
Sum-to-Product	$\cos x - \cos y = -2 \sin\left( \frac{x+y}{2} \right) \sin\left( \frac{x-y}{2} \right)$

Key Points and Applications

• Always arrange angles so that subtraction yields a positive angle when possible, to avoid negative angle complications.

• These formulas are useful for integrating trigonometric functions, simplifying expressions, and solving trigonometric equations.

• Exact values can often be found using the unit circle or trigonometric tables.

Additional info: These identities are especially useful in calculus, physics, and engineering for simplifying integrals and analyzing wave functions.