Sum to Product Formulas with Proof Derivation and Solved Examples
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:
\[{\bf{sin}}{\rm{ }}{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{sin}}{\rm{ }}{\bf{B}}{\rm{ }} = {\rm{ }}{\bf{2}}{\rm{ }}{\bf{sin}}\dfrac{{{\bf{A}} + {\bf{B}}}}{2}{\bf{cos}}\dfrac{{{\bf{A}} - B}}{2}\]
\[{\bf{sin}}{\rm{ }}{\bf{A}}{\rm{ }} - {\rm{ }}{\bf{sin}}{\rm{ }}{\bf{B}}{\rm{ }} = {\rm{ }}{\bf{2}}{\rm{ }}{\bf{sin}}\dfrac{{{\bf{A}} - {\bf{B}}}}{2}{\bf{cos}}\dfrac{{{\bf{A}} + {\bf{B}}}}{2}\]
\[{\bf{cos}}{\rm{ }}{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{cos}}{\rm{ }}{\bf{B}}{\rm{ }} = {\rm{ }}{\bf{2}}{\rm{ }}{\bf{cos}}\dfrac{{{\bf{A}} + {\bf{B}}}}{2}{\bf{cos}}\dfrac{{{\bf{A}} - {\bf{B}}}}{2}\]
\[{\bf{cosA}}{\rm{ }} - {\rm{ }}{\bf{cos}}{\rm{ }}{\bf{B}}{\rm{ }} = - {\rm{ }}{\bf{2}}{\rm{ }}{\bf{sin}}\dfrac{{{\bf{A}} + {\bf{B}}}}{2}{\bf{sin}}\dfrac{{{\bf{A}} - {\bf{B}}}}{2}\]
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:
\[{\bf{sin}}{\rm{ }}{\bf{A}}{\rm{ }}{\bf{cos}}{\rm{ }}{\bf{B}}{\rm{ }} = \dfrac{{\bf{1}}}{2}\left[ {{\rm{ }}{\bf{sin}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{B}}} \right){\rm{ }} + {\rm{ }}{\bf{sin}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} - {\rm{ }}{\bf{B}}} \right){\rm{ }}} \right]\]
\[{\bf{cos}}{\rm{ }}{\bf{A}}{\rm{ }}{\bf{sin}}{\rm{ }}{\bf{B}}{\rm{ }} = \dfrac{{\bf{1}}}{2}[{\rm{ }}{\bf{sin}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{B}}} \right){\rm{ }} - {\rm{ }}{\bf{sin}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} - {\rm{ }}{\bf{B}}} \right)]\]
\[{\bf{cos}}{\rm{ }}{\bf{A}}{\rm{ }}{\bf{cos}}{\rm{ }}{\bf{B}}{\rm{ }} = \dfrac{{\bf{1}}}{2}[{\bf{cos}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{B}}} \right){\rm{ }} + {\rm{ }}{\bf{cos}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} - {\rm{ }}{\bf{B}}} \right)]\]
\[{\bf{sin}}{\rm{ }}{\bf{A}}{\rm{ }}{\bf{sin}}{\rm{ }}{\bf{B}}{\rm{ }} = \dfrac{{\bf{1}}}{2}[{\bf{cos}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} - {\rm{ }}{\bf{B}}} \right){\rm{ }} - {\rm{ }}{\bf{cos}}{\rm{ }}\left( {{\bf{A}}{\rm{ }} + {\rm{ }}{\bf{B}}} \right)\;]\]
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.
FAQs on Sum To Product Formulas in Trigonometry
1. What are the sum to product formulas in trigonometry?
The sum to product formulas convert sums or differences of sine and cosine functions into products of trigonometric functions. The main identities are:
- sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2)
- sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2)
- cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2)
- cos A − cos B = −2 sin((A + B)/2) sin((A − B)/2)
2. How do you convert sin A + sin B into product form?
To convert sin A + sin B into product form, use the identity sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2). Steps:
- Add the angles: A + B
- Subtract the angles: A − B
- Divide each result by 2
- Substitute into the formula
- (50° + 30°)/2 = 40°
- (50° − 30°)/2 = 10°
3. What is the formula for cos A − cos B in product form?
The product form of cos A − cos B is −2 sin((A + B)/2) sin((A − B)/2). To apply it:
- Find (A + B)/2
- Find (A − B)/2
- Substitute into the formula with the negative sign
4. Why are sum to product identities useful?
Sum to product identities are useful because they simplify complex trigonometric expressions into easier product forms. They help in:
- Simplifying trigonometric equations
- Evaluating definite integrals
- Solving trigonometric equations
- Deriving other trigonometric identities
5. What is the difference between sum to product and product to sum formulas?
The difference is that sum to product formulas convert sums into products, while product to sum formulas convert products into sums. For example:
- Sum to product: sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2)
- Product to sum: 2 sin A cos B = sin(A + B) + sin(A − B)
6. How do you derive the sum to product identities?
Sum to product identities are derived using the angle addition and subtraction formulas for sine and cosine. For example:
- Start with sin(A + B) = sin A cos B + cos A sin B
- Also use sin(A − B) = sin A cos B − cos A sin B
- Add or subtract these equations
7. Can you give an example of using sum to product formulas?
Yes, for example, convert sin 80° − sin 20° into product form using sin A − sin B = 2 cos((A + B)/2) sin((A − B)/2). Steps:
- (80° + 20°)/2 = 50°
- (80° − 20°)/2 = 30°
8. Do sum to product formulas work for radians?
Yes, sum to product formulas work for both degrees and radians because they are trigonometric identities. For example, sin x + sin y = 2 sin((x + y)/2) cos((x − y)/2) holds whether x and y are measured in degrees or radians, as long as the unit is consistent.
9. How are sum to product identities used in integration?
Sum to product identities are used in integration to simplify sums of trigonometric functions into products that are easier to integrate. For example:
- ∫ (sin A + sin B) dx
- Rewrite as 2 sin((A + B)/2) cos((A − B)/2)
10. What are common mistakes when using sum to product formulas?
Common mistakes when using sum to product formulas include sign errors and incorrect angle division. Watch out for:
- Forgetting to divide (A + B) and (A − B) by 2
- Missing the negative sign in cos A − cos B
- Mixing degrees and radians
- Confusing sum to product with product to sum identities
