Product To Sum Formulas
The product to sum formulas are used to express the product of sine and cosine functions as a sum. These are derived from the sum and difference formulas of trigonometry. These formulas are very helpful while solving the integrals of trigonometric functions. Let us learn the product to sum formulas along with proofs and examples.
What Are Product To Sum Formulas?
The product-to-sum formulas are a set of formulas from trigonometric formulas and as we discussed in the previous section, they are derived from the sum and difference formulas. Here are the product to sum formulas and you can see their derivation below the formulas.
Product To Sum Formulas
There are 4 product to sum formulas that are widely used as trigonometric identities.
- sin A cos B = (1/2) [ sin (A + B) + sin (A - B) ]
- cos A sin B = (1/2) [ sin (A + B) - sin (A - B) ]
- cos A cos B = (1/2) [ cos (A + B) + cos (A - B) ]
- sin A sin B = (1/2) [ cos (A - B) - cos (A + B) ]
Product To Sum Formulas Derivation
We will use the sum/difference formulas of trigonometry to derive the product to sum formulas. Let us recall the sum and difference formulas of sin and cos and give some numbers to each of the sum/difference formulas.
sin (A + B) = sin A cos B + cos A sin B ... (1)
sin (A - B) = sin A cos B - cos A sin B ... (2)
cos (A + B) = cos A cos B - sin A sin B ... (3)
cos (A - B) = cos A cos B + sin A sin B ... (4)
Deriving the formula sin A cos B = (1/2) [ sin (A + B) + sin (A - B) ]:
Adding the equations (1) and (2), we get
sin (A + B) + sin (A - B) = 2 sin A cos B
Dividing both sides by 2,
sin A cos B = (1/2) [ sin (A + B) + sin (A - B) ]
Deriving the formula cos A sin B = (1/2) [ sin (A + B) - sin (A - B) ]:
Subtracting (2) from (1),
sin (A + B) - sin (A - B) = 2 cos A sin B
Dividing both sides by 2,
cos A sin B = (1/2) [ sin (A + B) - sin (A - B) ]
Deriving the formula cos A cos B = (1/2) [ cos (A + B) + cos (A - B) ]
Adding the equations (3) and (4), we get
cos (A + B) + cos (A - B) = 2 cos A cos B
Dividing both sides by 2,
cos A cos B = (1/2) [ cos (A + B) + cos (A - B) ]
Deriving the formula sin A sin B = (1/2) [ cos (A - B) - cos (A + B) ]
Subtracting (3) from (4),
cos (A - B) - cos (A + B) = 2 sin A sin B
Dividing both sides by 2,
sin A sin B = (1/2) [ cos (A - B) - cos (A + B) ]
You can see the applications of product to sum formulas in the section below.
Examples on Product To Sum Formulas
Example 1: Find the value of sin 75o sin 15o without actually evaluating the values of sin 75o and sin 15o.
Solution:
Using one of the product to sum formulas,
sin A sin B = (1/2) [ cos (A - B) - cos (A + B) ]
Substitute A = 75o and B = 15o, we get
sin 75o sin 15o = (1/2) [ cos (75o - 15o) - cos (75o + 15o) ]
= (1/2) [ cos 60o - cos 90o]
= (1/2) [ (1/2) - 0] (from trigonometry table)
= 1/4
Answer: sin 75o sin 15o = 1/4.
Example 2: Express 2 cos 5x sin 2x as sum/difference.
Solution:
Using one of the product to sum formulas,
cos A sin B = (1/2) [ sin (A + B) - sin (A - B) ]
Substitute A = 5x and B = 2x in the above formula,
cos 5x sin 2x = (1/2) [ sin (5x + 2x) - sin (5x - 2x) ]
cos 5x sin 2x = (1/2) [sin 7x - sin 3x]
Multiply both sides by 2,
2 cos 5x sin 2x = sin 7x - sin 3x
Answer: 2 cos 5x sin 2x = sin 7x - sin 3x.
Example 3: Find the value of the integral ∫ sin 3x cos 4x dx.
Solution:
Using one of the product to sum formulas,
sin A cos B = (1/2) [ sin (A + B) + sin (A - B) ]
Substitute A = 3x and B = 4x on both sides,
sin 3x cos 4x = (1/2) [ sin (3x + 4x) + sin (3x - 4x) ] = (1/2) [ sin 7x - sin x] (because sin (-x) = - sin x).
Now, we will evaluate the given integral using the above value.
∫ sin 3x cos 4x dx = ∫ (1/2) [ sin 7x - sin x] dx
= (1/2) [ -cos (7x) / 7 + cos x] + C (using integration by substitution)
Answer: ∫ sin 3x cos 4x dx = (1/2) [ -cos (7x) / 7 + cos x] + C.
FAQs on Product To Sum Formulas
What Are Product To Sum Formulas?
The product to sum formulas in trigonometry are formulas that are used to convert the product of trigonometric functions into the sum of trigonometric functions. There are 4 important product to sum formulas.
- sin A cos B = (1/2) [ sin (A + B) + sin (A - B) ]
- cos A sin B = (1/2) [ sin (A + B) - sin (A - B) ]
- cos A cos B = (1/2) [ cos (A + B) + cos (A - B) ]
- sin A sin B = (1/2) [ cos (A - B) - cos (A + B) ]
How To Derive Product To Sum Formulas?
The product to sum formulas are derived using the sum and difference formulas which are:
- sin (A + B) = sin A cos B + cos A sin B
- sin (A - B) = sin A cos B - cos A sin B
- cos (A + B) = cos A cos B - sin A sin B
- cos (A - B) = cos A cos B + sin A sin B
By adding or subtracting two of these four formulas, we can derive the product to sum formulas. You can find the detailed proof/derivation of product to sum formulas from the "What Are Product To Sum Formulas?" section of this page.
What Are the Applications of Product To Sum Formulas?
The product to sum formulas are used to write the product of two trigonometric functions (sin and cos) as the sum. These formulas are hence useful in integration as integrating a sum is pretty easier when compared to integrating a product.
How To Use Product To Sum Formulas?
Let us recall the product to sum formulas:
- sin A cos B = (1/2) [ sin (A + B) + sin (A - B) ]
- cos A sin B = (1/2) [ sin (A + B) - sin (A - B) ]
- cos A cos B = (1/2) [ cos (A + B) + cos (A - B) ]
- sin A sin B = (1/2) [ cos (A - B) - cos (A + B) ]
We can use one of these formulas to find the product of sin and (or) cos as the sum. For example if we have to convert cos 15o sin 45o into the sum, we will just apply the formula 2 from the above list. Then we get:
cos 15o sin 45o = (1/2) [ sin (15o + 45o) - sin (15o - 45o) ]
= (1/2) [ sin 60o + sin 30o] [because sin (-x) = - sin x]
= (1/2) [ (√3/2) + 1/2] (from trigonometry table)
= (√3 + 1) / 4