Express the Product as a Sum or Difference

We will how to express the product as a sum or difference.

1. Convert the product into sum or differences: 2 sin 5x cos 3x

Solution:

2 sin 5x cos 3x = sin (5x + 3x) + sin (5x -3x), [Since 2 sin A cos B = sin (A + B) + sin (A - B)]

= sin 8x + sin 2x


2. Express sin (3∅)/2 ∙ cos (5∅)/2 as sum or difference. 

Solution:

sin (3∅)/2  cos (5∅)/2  

= 1/2 ∙ 2sin (3∅)/2 cos (5∅)/2

 = 1/2 [sin ((3∅)/2 + (5∅)/2) - sin ((5∅)/2 - (3∅)/2)]

= 1/2 (sin 4∅ - sin ∅)

3. Convert 2 cos 5α sin 3α into sum or differences.

Solution:

2 cos 5α sin 3α = sin (5α + 3α) - sin (5α -3α), [Since 2 cos A sin B = sin (A + B) - sin (A - B)]

= sin 8α - sin 2α

 

4. Express the product as a sum or difference: 4 sin 20° sin 35°

Solution:

4sin 20° sin 35° = 2 ∙ 2 sin20° sin 35°

= 2 [cos (35°- 20°) - cos (35° + 20°)]

= 2 (cos 15° - cos 55°).

 

5. Convert  cos 9β cos 4β into sum or differences.

Solution:

cos 9β cos 4β = ½ ∙ 2 cos 9β cos 4β

= ½ [cos (9β + 4β) + cos (9β - 4β)], [Since 2 cos A cos B = cos (A + B) + cos (A - B)]

= ½ (cos 13β + cos 5β)

 

6. Prove that, tan (60° - ∅) tan (60° + ∅) = (2 cos 2∅ + 1)/(2 cos 2∅ - 1)

Solution:

L.H.S. = tan (60° - ∅) tan (60° + ∅)

         = (2 sin (60° - ∅) sin (60° + ∅))/(2cos (60° - ∅) cos (60° + ∅)

         = cos [(60° + ∅) - (60° - ∅)] - cos [(60° + ∅)+ (60° - ∅) ]/(cos[(60° + ∅ )+ (60° - ∅) ] + cos [(60° + ∅) - (60° - ∅) ] )

         = (cos 2∅ - cos 120°)/(cos 120° + cos 2∅)

         = (cos 2∅ - (-1/2))/(-1/2 + cos 2∅), [Since cos 120° = -1/2]

         = (cos 2∅ + 1/2)/(cos 2∅ - 1/2)

         = (2 cos 2∅ + 1)/(2 cos 2∅ - 1)   proved

 

7. Convert the product into sum or differences: 3 sin 13β sin 3β

Solution:

3 sin 13β sin 3β = 3/2 ∙ 2 sin 13β sin 3β

= 3/2 [cos (13β - 3β) - cos (13β + 3β)], [Since 2 sin A sin B = cos (A - B) - cos (A + B)]

= 3/2 (cos 10β - cos 16β)

 

8. Show that, 4 sin A sin B sin C = sin (A + B - C) + sin (B + C - A) + sin (C+ A - B) - sin (A + B + C)

Solution:

L.H.S. = 4 sin A sin B sin C

= 2 sin A (2 sin B sin C)

= 2 sin A {cos (B - C) - cos (B + C)}

= 2 sin A ∙ cos (B - C) - 2 sin A cos (B + C)

= sin (A + B - C) + sin (A - B + C) - [sin (A + B + C) - sin (B + C -A)]

= sin (A + B - C) + sin (B + C - A) + sin (A + C - B) - sin (A + B + C) = R.H.S.

                                                                                                      Proved

 Converting Product into Sum/Difference and Vice Versa






11 and 12 Grade Math

From Express the Product as a Sum or Difference to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Comparing and Ordering Decimals |Arranging Decimals

    Apr 19, 25 12:16 PM

    Arranging Decimals
    Practice different types of math questions given in the worksheet on comparing and ordering decimals. This worksheet contains questions mainly related to compare decimals and then place the decimals i…

    Read More

  2. Comparison of Decimal Fractions | Comparing Decimals Numbers | Decimal

    Apr 19, 25 11:47 AM

    Comparison of Decimal Fractions
    While comparing natural numbers we first compare total number of digits in both the numbers and if they are equal then we compare the digit at the extreme left. If they also equal then we compare the…

    Read More

  3. Expanded form of Decimal Fractions |How to Write a Decimal in Expanded

    Apr 19, 25 11:25 AM

    Expanded form of Decimal
    Decimal numbers can be expressed in expanded form using the place-value chart. In expanded form of decimal fractions we will learn how to read and write the decimal numbers. Note: When a decimal is mi…

    Read More

  4. Missing Numbers up to 10 | Worksheets on Missing Numbers up to 10

    Apr 18, 25 04:53 PM

    missing numbers up to 10
    Printable worksheets on missing numbers up to 10 help the kids to practice counting of the numbers.

    Read More

  5. Ordering Decimals | Comparing Decimals | Ascending & Descending Order

    Apr 18, 25 01:49 PM

    Ordering Decimal Numbers
    In ordering decimals we will learn how to compare two or more decimals. (i) Convert each of them as like decimals. (ii) Compare these decimals just as we compare two whole numbers ignoring

    Read More