Sum and Product Formulae in Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. The sum and product formulae are important tools in trigonometry that allow us to express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sines and cosines of the individual angles. These formulae are essential for simplifying complex trigonometric expressions and solving trigonometric equations.

Sum and Difference Formulas

The sum and difference formulas for sine, cosine, and tangent are as follows:

Sine

For any angles $A$ and $B$:

$$ \sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B) $$

Cosine

For any angles $A$ and $B$:

$$ \cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B) $$

Tangent

For any angles $A$ and $B$, provided that $\cos(A) \neq 0$ and $\cos(B) \neq 0$:

$$ \tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)} $$

Product-to-Sum Formulas

The product-to-sum formulas are derived from the sum and difference formulas and are used to express products of sines and cosines as sums or differences. They are as follows:

Sine and Cosine

$$ \sin(A)\cos(B) = \frac{1}{2}[\sin(A + B) + \sin(A - B)] $$

Sine and Sine

$$ \sin(A)\sin(B) = \frac{1}{2}[\cos(A - B) - \cos(A + B)] $$

Cosine and Cosine

$$ \cos(A)\cos(B) = \frac{1}{2}[\cos(A + B) + \cos(A - B)] $$

Sum-to-Product Formulas

Conversely, the sum-to-product formulas are used to express sums or differences of sines and cosines as products. They are as follows:

Sine and Sine

$$ \sin(A) \pm \sin(B) = 2\sin\left(\frac{A \pm B}{2}\right)\cos\left(\frac{A \mp B}{2}\right) $$

Cosine and Cosine

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

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

Examples

Let's go through some examples to understand how to apply these formulas.

Example 1: Sum Formula for Sine

Calculate $\sin(75^\circ)$ using the sum formula for sine.

$$ \sin(75^\circ) = \sin(45^\circ + 30^\circ) $$

Using the sum formula:

$$ \sin(75^\circ) = \sin(45^\circ)\cos(30^\circ) + \cos(45^\circ)\sin(30^\circ) $$

$$ \sin(75^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} $$

$$ \sin(75^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4} $$

Example 2: Product-to-Sum Formula

Express $\sin(x)\cos(y)$ as a sum.

Using the product-to-sum formula:

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

Example 3: Sum-to-Product Formula

Express $\sin(x) + \sin(y)$ as a product.

Using the sum-to-product formula:

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

Table of Differences and Important Points

Property	Sum Formulas	Product-to-Sum Formulas	Sum-to-Product Formulas
Sine	$\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$	$\sin(A)\cos(B) = \frac{1}{2}[\sin(A + B) + \sin(A - B)]$	$\sin(A) \pm \sin(B) = 2\sin\left(\frac{A \pm B}{2}\right)\cos\left(\frac{A \mp B}{2}\right)$
Cosine	$\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$	$\cos(A)\cos(B) = \frac{1}{2}[\cos(A + B) + \cos(A - B)]$	$\cos(A) + \cos(B) = 2\cos\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right)$
Tangent	$\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}$	N/A	N/A
Application	Simplify expressions involving the sum or difference of angles	Convert products of trigonometric functions into sums	Convert sums of trigonometric functions into products
Prerequisite	Knowledge of individual sine and cosine values	Knowledge of sum and difference formulas	Knowledge of sum and difference formulas

Understanding and applying these formulas is crucial for solving trigonometric problems, especially in calculus, physics, and engineering. Practice using these formulas with various angles to become proficient in their application.