# Product To Sum Formula

## Product to Sum Formulas

The sum of the sine and cosine products is expressed using the Product to Sum Formula. These are obtained from the trigonometric sum and difference formulae. When attempting to solve the integrals of trigonometric functions, these formulae are highly beneficial. Students can study the proofs and examples for the Product to Sum Formula.

The notes and solutions of the Extramarks’ Product to Sum Formula are also available in Hindi, making it easier for students of other boards to comprehend these concepts better. Experts have made sure that the notes and solutions for the Product to Sum Formula are in accordance with the NCERT syllabus and CBSE norms, while also emulating the structure of the NCERT books.

### What Are Product to Sum Formulas?

As we discussed in the previous section, the sum and difference formulas are the source of the Product to Sum Formula, a collection of trigonometric formulas. On the Extramarks website and mobile application, the Product to Sum Formula is referenced, and students can view their derivation below the formulas.

The Extramarks notes solutions for the Product to Sum Formula have been designed and prepared after considering and making sure that they cater fully to all student needs while also considering the pattern of the past years’ question papers, making it more convenient for students to understand what kind of questions might come in the examination.

### Product to Sum Formulas

The Product to Sum Formula is derived by students using the trigonometry sum/difference formulas. With the use of Extramarks’ Product to Sum Formula and offering some numbers to each of the sum/difference formulae, students may recall the sum and difference formulas of sin and cos.

• 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)

The notes and solutions of Extramarks’ Product to Sum Formula are extremely versatile, meaning students of all boards can refer to them since they are available in Hindi as well, thereby disrupting the language barrier.

These notes on the Product to Sum Formula are also extremely student-friendly. They are not very difficult to comprehend while also being difficult enough and not lowering the standard of learning of students.

### Product to Sum Formula Derivation

Below are the steps and explanation of how the aforementioned formulas have been derived for better student understanding:

• Constructing the equation 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)]

• Constructing 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)]

• Constructing 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)]

• Constructing 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)]

Extramarks experts have made sure to design and curate the solutions for Product to Sum Formula as such that the framework of the solutions are easy to understand and comprehend, thereby making sure that students face no difficulty while practising the sums.

Extramarks’ notes and solutions on the Product to Sum Formula are also dynamic, diverse and varied, meaning, experts always stay updated on the new norms and regulations put in place by the CBSE board and pertain to them thereby always providing accurate answers.

The notes for Product to Sum Formula by Extramarks also make sure that they cater to all the needs of each individual student, making it extremely student-friendly.

### Examples on Product to Sum Formula

Example 1: Calculate the value of sin 75o and sin 15o without actually assessing their values.

Solution:

Sin A sin B = (1/2) using one of the Product to Sum Formula. [cos(A-B) – cos(A+B)]

If we replace A with 75° and B with 15°, we obtain sin 75° sin 15°. = (1/2) (75o – 15o) – (75o + 15o) cos

= (1/2) (1/2) = [cos 60o – cos 90o] [(1/2) – 0] 1/4 (from a trigonometry table)

Answer: sin 75o sin 15o = 1/4.

Example 2: Write the sum or difference as 2 cos 5x sin 2x.

Solution:

Cos A sin B = (1/2) using one of the Product to Sum Formula. (A + B) sin – (A – B) sin

The formula above, cos 5x sin 2x = (1/2), should be changed to A = 5x and B = 2x. Sin (5x + 2x) − Sin (5x – 2x)

sin 2x cos 5 = (1/2) [7x sin – 3x sin]

Divide both sides by 2, 2 cos 5x sin 2x, which equals sin 7x − sin 3x.

The solution is sin 7x – sin 3x = 2 cos 5x sin 2x.

Example 3: Calculate the integral’s value, sin 3x cos 4x dx.

Solution:

Sin A cos B = (1/2) using one of the Product to Sum Formula. Sin (A + B) + Sin (A – B)

Replace A with (3x) and B with (4x) such that sin (3x) cos(4x) = (1/2). [sin (3x+4) + sin (3x-4)] = [sin 7x – sin x] (1/2) (because sin(-x) = sin(-x)).

We will now use the aforementioned value to evaluate the provided integral.

(1/2) = sin 3x cos 4x dx Sin x – 7x sin dx\s= (1/2) [7+cos x] / [-cos (7x)] + C (using integration by substitution) (using integration by substitution)

Sin 3x cos 4x dx = (1/2) is the solution. [7+cos x] / [-cos (7x)] + C.

The examples above will give students a better and clear idea of solving the sums in their final examinations. The examples have been compiled in collaboration with the top experts at Extramarks to provide students with all they need and all they will ever need under one roof in one place.