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. Successor and Predecessor | Successor of a Whole Number | Predecessor

    Jul 29, 25 12:59 AM

    Successor and Predecessor
    The number that comes just before a number is called the predecessor. So, the predecessor of a given number is 1 less than the given number. Successor of a given number is 1 more than the given number…

    Read More

  2. Worksheet on Area, Perimeter and Volume | Square, Rectangle, Cube,Cubo

    Jul 28, 25 01:52 PM

    Volume of a Cuboids
    In this worksheet on area perimeter and volume you will get different types of questions on find the perimeter of a rectangle, find the perimeter of a square, find the area of a rectangle, find the ar…

    Read More

  3. Worksheet on Volume of a Cube and Cuboid |The Volume of a RectangleBox

    Jul 25, 25 03:15 AM

    Volume of a Cube and Cuboid
    We will practice the questions given in the worksheet on volume of a cube and cuboid. We know the volume of an object is the amount of space occupied by the object.1. Fill in the blanks:

    Read More

  4. Volume of a Cuboid | Volume of Cuboid Formula | How to Find the Volume

    Jul 24, 25 03:46 PM

    Volume of Cuboid
    Cuboid is a solid box whose every surface is a rectangle of same area or different areas. A cuboid will have a length, breadth and height. Hence we can conclude that volume is 3 dimensional. To measur…

    Read More

  5. Volume of a Cube | How to Calculate the Volume of a Cube? | Examples

    Jul 23, 25 11:37 AM

    Volume of a Cube
    A cube is a solid box whose every surface is a square of same area. Take an empty box with open top in the shape of a cube whose each edge is 2 cm. Now fit cubes of edges 1 cm in it. From the figure i…

    Read More