Sum to Product Formulas

In trigonometry, the sum-to-product formula is a set of formulas that are used to express the sine and cosine functions' sum, difference, and products. These formulas allow us to express the sine and cosine trigonometric functions as the sum or difference of their products, which simplifies mathematical problems. By substituting the variables, it is possible to derive the sum-to-product formulas from the product-to-sum formulas in trigonometry.

In this article, we will thoroughly examine the sum-to-product formula and show how to obtain it from the product-to-sum formulas.

What are the Sum-to-Product Formulas?

The Sum-to-product formula is used to express the sum or difference of a sine function and the sum or difference of a cosine function as the product of a sine and cosine function. By applying this formula, the sum or difference of trigonometric functions of sine and cosine can be expressed as a product, simplifying mathematical problems. The product for sum formula in trigonometry can be used to obtain the sum for product formula with a variable change.

List of the Sum-to-Product Formula

Following is the list of the sum-to-product formula:

1. \[{\bf{sin}}{\rm{ }}{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{sin}}{\rm{ }}{\bf{B}}{\rm{ }} = {\rm{ }}{\bf{2}}{\rm{ }}{\bf{sin}}\dfrac{{{\bf{A}} + {\bf{B}}}}{2}{\bf{cos}}\dfrac{{{\bf{A}} - B}}{2}\]

2. \[{\bf{sin}}{\rm{ }}{\bf{A}}{\rm{ }} - {\rm{ }}{\bf{sin}}{\rm{ }}{\bf{B}}{\rm{ }} = {\rm{ }}{\bf{2}}{\rm{ }}{\bf{sin}}\dfrac{{{\bf{A}} - {\bf{B}}}}{2}{\bf{cos}}\dfrac{{{\bf{A}} + {\bf{B}}}}{2}\]

3. \[{\bf{cos}}{\rm{ }}{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{cos}}{\rm{ }}{\bf{B}}{\rm{ }} = {\rm{ }}{\bf{2}}{\rm{ }}{\bf{cos}}\dfrac{{{\bf{A}} + {\bf{B}}}}{2}{\bf{cos}}\dfrac{{{\bf{A}} - {\bf{B}}}}{2}\]

4. \[{\bf{cosA}}{\rm{ }} - {\rm{ }}{\bf{cos}}{\rm{ }}{\bf{B}}{\rm{ }} = - {\rm{ }}{\bf{2}}{\rm{ }}{\bf{sin}}\dfrac{{{\bf{A}} + {\bf{B}}}}{2}{\bf{sin}}\dfrac{{{\bf{A}} - {\bf{B}}}}{2}\]

What is the Product to-Sum Formula?

The product-to-sum formula is used to express the product of sine and cosine functions as a sum. They are derived from the trigonometric sum and difference formulas. This formula is very useful for solving integrals of trigonometric functions.

List of Product-to-Sum Formula

Following is the list of product to sum formula:

1. \[{\bf{sin}}{\rm{ }}{\bf{A}}{\rm{ }}{\bf{cos}}{\rm{ }}{\bf{B}}{\rm{ }} = \dfrac{{\bf{1}}}{2}\left[ {{\rm{ }}{\bf{sin}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{B}}} \right){\rm{ }} + {\rm{ }}{\bf{sin}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} - {\rm{ }}{\bf{B}}} \right){\rm{ }}} \right]\]

2. \[{\bf{cos}}{\rm{ }}{\bf{A}}{\rm{ }}{\bf{sin}}{\rm{ }}{\bf{B}}{\rm{ }} = \dfrac{{\bf{1}}}{2}[{\rm{ }}{\bf{sin}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{B}}} \right){\rm{ }} - {\rm{ }}{\bf{sin}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} - {\rm{ }}{\bf{B}}} \right)]\]

3. \[{\bf{cos}}{\rm{ }}{\bf{A}}{\rm{ }}{\bf{cos}}{\rm{ }}{\bf{B}}{\rm{ }} = \dfrac{{\bf{1}}}{2}[{\bf{cos}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{B}}} \right){\rm{ }} + {\rm{ }}{\bf{cos}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} - {\rm{ }}{\bf{B}}} \right)]\]

4. \[{\bf{sin}}{\rm{ }}{\bf{A}}{\rm{ }}{\bf{sin}}{\rm{ }}{\bf{B}}{\rm{ }} = \dfrac{{\bf{1}}}{2}[{\bf{cos}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} - {\rm{ }}{\bf{B}}} \right){\rm{ }} - {\rm{ }}{\bf{cos}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{B}}} \right)\;]\]

Factorization and Defactorization of Formula

Factorization formulas are those Sum-to-product formulas used to express the sum or difference of a sine function and the sum or difference of a cosine function as the product of a sine and cosine function. The sum to product formula is known as the factorization formula. Factorization formulas are those product-to-sum formulas that are used to express the product of sine and cosine functions as a sum. Product-to-sum formulas are known as factorization formulas.

Important Points for Sum-to-Product Formula

• The Sum to product formula is used to express the sum and difference of trigonometric functions of sine and cosine as the product of sine and cosine functions.

• You can use the sum-to-product formula of trigonometry to derive the sum formula as a product.

• You can apply these formulas to simplify trigonometry problems.
  

Solved Examples

1.Express\[\;{\bf{cos4x}} + {\bf{cos2x}}\] as a product.

Solution: Using sum to product formula of \[{\bf{cosA}} + {\bf{cosB}} = {\bf{2}}{\rm{ }}{\bf{cos}}\dfrac{{{\bf{A}} + {\bf{B}}}}{2}{\bf{cos}}\dfrac{{{\bf{A}} - {\bf{B}}}}{2}\]

Here \[\begin{array}{*{20}{l}}{{\bf{A}} = {\bf{4x}}{\rm{ }},{\rm{ }}{\bf{B}} = {\bf{2x}}}\\{{\bf{cos4x}} + {\bf{cos2x}} = {\bf{2}}{\rm{ }}{\bf{cos}}\dfrac{{{\bf{4x}} + {\bf{2x}}}}{2}{\bf{cos}}\dfrac{{{\bf{4x}} - {\bf{2x}}}}{2}}\\\begin{array}{l}{\bf{cos4x}} + {\bf{cos2x}} = {\bf{2}}{\rm{ }}{\bf{cos}}\dfrac{{{\bf{6x}}}}{2}{\bf{cos}}\dfrac{{{\bf{2x}}}}{2}\\{\bf{cos4x}} + {\bf{cos2x}} = {\bf{2}}{\rm{ }}{\bf{cos}}{\rm{ }}{\bf{3x}}{\rm{ }}{\bf{cos}}{\rm{ }}{\bf{x}}\end{array}\end{array}\]

2.Express \[{\bf{sin9x}} - {\bf{sin2x}}\] as a product.

Solution: Using sum to product formula of \[{\bf{sinA}} - {\bf{sinB}} = {\bf{2}}{\rm{ }}{\bf{sin}}\dfrac{{{\bf{A}} - {\bf{B}}}}{2}{\bf{cos}}\dfrac{{{\bf{A}} + {\bf{B}}}}{2}\]

Here \[\begin{array}{*{20}{l}}{{\bf{A}} = {\bf{9x}}{\rm{ }},{\rm{ }}{\bf{B}} = {\bf{2x}}}\\\begin{array}{l}{\bf{sin9x}} - {\bf{sin2x}} = {\bf{2}}{\rm{ }}{\bf{sin}}\dfrac{{{\bf{9x}} - {\bf{2x}}}}{2}{\bf{cos}}\dfrac{{{\bf{9x}} - {\bf{2x}}}}{2}\\{\bf{sin9x}} - {\bf{sin2x}} = {\bf{2}}{\rm{ }}{\bf{sin}}\dfrac{{{\bf{7x}}}}{2}{\bf{cos}}\dfrac{{{\bf{11x}}}}{2}\end{array}\end{array}\]

3.Prove \[\dfrac{{\left( {{\bf{sin4x}} - {\bf{sin2x}}} \right)}}{{{\bf{cos3x}}}} = {\bf{2sinx}}\]

Solution: L.H.S. \[\;\dfrac{{\left( {{\bf{sin4x}} - {\bf{sin2x}}} \right)}}{{{\bf{cos3x}}}}\]

Using sum to product formula of \[\;{\bf{sinA}} - {\bf{sinB}} = {\bf{2}}{\rm{ }}{\bf{sin}}\dfrac{{{\bf{A}} - {\bf{B}}}}{2}{\bf{cos}}\dfrac{{{\bf{A}} + {\bf{B}}}}{2}\]

Here \[\begin{array}{*{20}{l}}{{\bf{A}} = {\bf{4x}}{\rm{ }},{\rm{ }}{\bf{B}} = {\bf{2x}}}\\\begin{array}{l}{\bf{sin4x}} - {\bf{sin2x}} = {\bf{2}}{\rm{ }}{\bf{sin}}\dfrac{{{\bf{4x}} - {\bf{2x}}}}{2}{\bf{cos}}\dfrac{{{\bf{4x}} + {\bf{2x}}}}{2}\\{\bf{sin9x}} - {\bf{sin2x}} = {\bf{2}}{\rm{ }}{\bf{sin}}{\rm{ }}{\bf{x}}{\rm{ }}{\bf{cos}}{\rm{ }}{\bf{3x}}\end{array}\end{array}\]

Now,

\[\begin{array}{l}\;\dfrac{{\left( {{\bf{sin4x}} - {\bf{sin2x}}} \right)}}{{{\bf{cos3x}}}} = \dfrac{{{\bf{2}}{\rm{ }}{\bf{sinx}}{\rm{ }}{\bf{cos3x}}}}{{{\bf{cos3x}}}}\\\; \Rightarrow \dfrac{{\left( {{\bf{sin4x}} - {\bf{sin2x}}} \right)}}{{{\bf{cos3x}}}} = {\bf{2sinx}} = R.H.S.\end{array}\]

Hence proved.

Conclusion

The sum-to-product formula in trigonometry is a set of formulas that are used to express the sum, difference, and products of the sine and cosine functions. The sum-to-product formulas in trigonometry can be obtained by substituting the variables in the product-to-sum formulas.