7.4 Sum-to-Product and Product-to-Sum Formulas - Precalculus 2e | OpenStax (2024)

Learning Objectives

In this section, you will:

  • Express products as sums.
  • Express sums as products.
7.4 Sum-to-Product and Product-to-Sum Formulas - Precalculus 2e | OpenStax (1)

Figure 1 The UCLA marching band (credit: Eric Chan, Flickr).

A band marches down the field creating an amazing sound that bolsters the crowd. That sound travels as a wave that can be interpreted using trigonometric functions. For example, Figure 2 represents a sound wave for the musical note A. In this section, we will investigate trigonometric identities that are the foundation of everyday phenomena such as sound waves.

7.4 Sum-to-Product and Product-to-Sum Formulas - Precalculus 2e | OpenStax (2)

Figure 2

Expressing Products as Sums

We have already learned a number of formulas useful for expanding or simplifying trigonometric expressions, but sometimes we may need to express the product of cosine and sine as a sum. We can use the product-to-sum formulas, which express products of trigonometric functions as sums. Let’s investigate the cosine identity first and then the sine identity.

Expressing Products as Sums for Cosine

We can derive the product-to-sum formula from the sum and difference identities for cosine. If we add the two equations, we get:

cosαcosβ+sinαsinβ=cos( αβ ) +cosαcosβsinαsinβ=cos( α+β ) ________________________________ 2cosαcosβ=cos( αβ )+cos( α+β ) cosαcosβ+sinαsinβ=cos( αβ ) +cosαcosβsinαsinβ=cos( α+β ) ________________________________ 2cosαcosβ=cos( αβ )+cos( α+β )

Then, we divide by 2 2 to isolate the product of cosines:

cosαcosβ= 1 2 [cos(αβ)+cos(α+β)] cosαcosβ= 1 2 [cos(αβ)+cos(α+β)]

How To

Given a product of cosines, express as a sum.

  1. Write the formula for the product of cosines.
  2. Substitute the given angles into the formula.
  3. Simplify.

Example 1

Writing the Product as a Sum Using the Product-to-Sum Formula for Cosine

Write the following product of cosines as a sum: 2cos( 7x 2 )cos 3x 2 . 2cos( 7x 2 )cos 3x 2 .

Solution

We begin by writing the formula for the product of cosines:

cosαcosβ= 1 2 [ cos( αβ )+cos( α+β ) ] cosαcosβ= 1 2 [ cos( αβ )+cos( α+β ) ]

We can then substitute the given angles into the formula and simplify.

2cos( 7x 2 )cos( 3x 2 )=(2)( 1 2 )[ cos( 7x 2 3x 2 )+cos( 7x 2 + 3x 2 ) ] =[ cos( 4x 2 )+cos( 10x 2 ) ] =cos2x+cos5x 2cos( 7x 2 )cos( 3x 2 )=(2)( 1 2 )[ cos( 7x 2 3x 2 )+cos( 7x 2 + 3x 2 ) ] =[ cos( 4x 2 )+cos( 10x 2 ) ] =cos2x+cos5x

Try It #1

Use the product-to-sum formula to write the product as a sum or difference: cos( 2θ )cos( 4θ ). cos( 2θ )cos( 4θ ).

Expressing the Product of Sine and Cosine as a Sum

Next, we will derive the product-to-sum formula for sine and cosine from the sum and difference formulas for sine. If we add the sum and difference identities, we get:

sin(α+β)=sinαcosβ+cosαsinβ +sin(αβ)=sinαcosβcosαsinβ _________________________________________ sin(α+β)+sin(αβ)=2sinαcosβ sin(α+β)=sinαcosβ+cosαsinβ +sin(αβ)=sinαcosβcosαsinβ _________________________________________ sin(α+β)+sin(αβ)=2sinαcosβ

Then, we divide by 2 to isolate the product of cosine and sine:

sinαcosβ= 1 2 [ sin( α+β )+sin( αβ ) ] sinαcosβ= 1 2 [ sin( α+β )+sin( αβ ) ]

Example 2

Writing the Product as a Sum Containing only Sine or Cosine

Express the following product as a sum containing only sine or cosine and no products: sin( 4θ )cos( 2θ ). sin( 4θ )cos( 2θ ).

Solution

Write the formula for the product of sine and cosine. Then substitute the given values into the formula and simplify.

sinαcosβ= 1 2 [ sin( α+β )+sin( αβ ) ] sin( 4θ )cos( 2θ )= 1 2 [ sin( 4θ+2θ )+sin( 4θ2θ ) ] = 1 2 [ sin( 6θ )+sin( 2θ ) ] sinαcosβ= 1 2 [ sin( α+β )+sin( αβ ) ] sin( 4θ )cos( 2θ )= 1 2 [ sin( 4θ+2θ )+sin( 4θ2θ ) ] = 1 2 [ sin( 6θ )+sin( 2θ ) ]

Try It #2

Use the product-to-sum formula to write the product as a sum: sin( x+y )cos( xy ). sin( x+y )cos( xy ).

Expressing Products of Sines in Terms of Cosine

Expressing the product of sines in terms of cosine is also derived from the sum and difference identities for cosine. In this case, we will first subtract the two cosine formulas:

cos( αβ )=cosαcosβ+sinαsinβ cos( α+β )=( cosαcosβsinαsinβ ) ____________________________________________________ cos( αβ )cos( α+β )=2sinαsinβ cos( αβ )=cosαcosβ+sinαsinβ cos( α+β )=( cosαcosβsinαsinβ ) ____________________________________________________ cos( αβ )cos( α+β )=2sinαsinβ

Then, we divide by 2 to isolate the product of sines:

sinαsinβ= 1 2 [ cos( αβ )cos( α+β ) ] sinαsinβ= 1 2 [ cos( αβ )cos( α+β ) ]

Similarly we could express the product of cosines in terms of sine or derive other product-to-sum formulas.

The Product-to-Sum Formulas

The product-to-sum formulas are as follows:

cosαcosβ= 1 2 [ cos( αβ )+cos( α+β ) ] cosαcosβ= 1 2 [ cos( αβ )+cos( α+β ) ]

sinαcosβ= 1 2 [ sin( α+β )+sin( αβ ) ] sinαcosβ= 1 2 [ sin( α+β )+sin( αβ ) ]

sinαsinβ= 1 2 [ cos( αβ )cos( α+β ) ] sinαsinβ= 1 2 [ cos( αβ )cos( α+β ) ]

cosαsinβ= 1 2 [ sin( α+β )sin( αβ ) ] cosαsinβ= 1 2 [ sin( α+β )sin( αβ ) ]

Example 3

Express the Product as a Sum or Difference

Write cos(3θ)cos(5θ) cos(3θ)cos(5θ) as a sum or difference.

Solution

We have the product of cosines, so we begin by writing the related formula. Then we substitute the given angles and simplify.

cosαcosβ= 1 2 [cos(αβ)+cos(α+β)] cos(3θ)cos(5θ)= 1 2 [cos(3θ5θ)+cos(3θ+5θ)] = 1 2 [cos(2θ)+cos(8θ)] Useeven-oddidentity. cosαcosβ= 1 2 [cos(αβ)+cos(α+β)] cos(3θ)cos(5θ)= 1 2 [cos(3θ5θ)+cos(3θ+5θ)] = 1 2 [cos(2θ)+cos(8θ)] Useeven-oddidentity.

Try It #3

Use the product-to-sum formula to evaluate cos 11π 12 cos π 12 . cos 11π 12 cos π 12 .

Expressing Sums as Products

Some problems require the reverse of the process we just used. The sum-to-product formulas allow us to express sums of sine or cosine as products. These formulas can be derived from the product-to-sum identities. For example, with a few substitutions, we can derive the sum-to-product identity for sine. Let u+v 2 =α u+v 2 =α and uv 2 =β. uv 2 =β.

Then,

α+β= u+v 2 + uv 2 = 2u 2 =u αβ= u+v 2 uv 2 = 2v 2 =v α+β= u+v 2 + uv 2 = 2u 2 =u αβ= u+v 2 uv 2 = 2v 2 =v

Thus, replacing α α and β β in the product-to-sum formula with the substitute expressions, we have

sinαcosβ= 1 2 [sin(α+β)+sin(αβ)] sin( u+v 2 )cos( uv 2 )= 1 2 [sinu+sinv] Substitutefor(α+β)and(αβ) 2sin( u+v 2 )cos( uv 2 )=sinu+sinv sinαcosβ= 1 2 [sin(α+β)+sin(αβ)] sin( u+v 2 )cos( uv 2 )= 1 2 [sinu+sinv] Substitutefor(α+β)and(αβ) 2sin( u+v 2 )cos( uv 2 )=sinu+sinv

The other sum-to-product identities are derived similarly.

Sum-to-Product Formulas

The sum-to-product formulas are as follows:

sinα+sinβ=2sin( α+β 2 )cos( αβ 2 ) sinα+sinβ=2sin( α+β 2 )cos( αβ 2 )

sinαsinβ=2sin( αβ 2 )cos( α+β 2 ) sinαsinβ=2sin( αβ 2 )cos( α+β 2 )

cosαcosβ=2sin( α+β 2 )sin( αβ 2 ) cosαcosβ=2sin( α+β 2 )sin( αβ 2 )

cosα+cosβ=2cos( α+β 2 )cos( αβ 2 ) cosα+cosβ=2cos( α+β 2 )cos( αβ 2 )

Example 4

Writing the Difference of Sines as a Product

Write the following difference of sines expression as a product: sin( 4θ )sin( 2θ ). sin( 4θ )sin( 2θ ).

Solution

We begin by writing the formula for the difference of sines.

sinαsinβ=2sin( αβ 2 )cos( α+β 2 ) sinαsinβ=2sin( αβ 2 )cos( α+β 2 )

Substitute the values into the formula, and simplify.

sin(4θ)sin(2θ)=2sin( 4θ2θ 2 )cos( 4θ+2θ 2 ) =2sin( 2θ 2 )cos( 6θ 2 ) =2sinθcos(3θ) sin(4θ)sin(2θ)=2sin( 4θ2θ 2 )cos( 4θ+2θ 2 ) =2sin( 2θ 2 )cos( 6θ 2 ) =2sinθcos(3θ)

Try It #4

Use the sum-to-product formula to write the sum as a product: sin( 3θ )+sin( θ ). sin( 3θ )+sin( θ ).

Example 5

Evaluating Using the Sum-to-Product Formula

Evaluate cos( 15 )cos( 75 ). cos( 15 )cos( 75 ).

Solution

We begin by writing the formula for the difference of cosines.

cosαcosβ=2sin( α+β 2 )sin( αβ 2 ) cosαcosβ=2sin( α+β 2 )sin( αβ 2 )

Then we substitute the given angles and simplify.

cos( 15 )cos( 75 )=2sin( 15 + 75 2 )sin( 15 75 2 ) =2sin( 45 )sin( 30 ) =2( 2 2 )( 1 2 ) = 2 2 cos( 15 )cos( 75 )=2sin( 15 + 75 2 )sin( 15 75 2 ) =2sin( 45 )sin( 30 ) =2( 2 2 )( 1 2 ) = 2 2

Example 6

Proving an Identity

Prove the identity:

cos( 4t )cos( 2t ) sin( 4t )+sin( 2t ) =tant cos( 4t )cos( 2t ) sin( 4t )+sin( 2t ) =tant

Solution

We will start with the left side, the more complicated side of the equation, and rewrite the expression until it matches the right side.

cos(4t)cos(2t) sin(4t)+sin(2t) = 2sin( 4t+2t 2 )sin( 4t2t 2 ) 2sin( 4t+2t 2 )cos( 4t2t 2 ) = 2sin(3t)sint 2sin(3t)cost = 2 sin(3t) sint 2 sin(3t) cost = sint cost =tant cos(4t)cos(2t) sin(4t)+sin(2t) = 2sin( 4t+2t 2 )sin( 4t2t 2 ) 2sin( 4t+2t 2 )cos( 4t2t 2 ) = 2sin(3t)sint 2sin(3t)cost = 2 sin(3t) sint 2 sin(3t) cost = sint cost =tant

Analysis

Recall that verifying trigonometric identities has its own set of rules. The procedures for solving an equation are not the same as the procedures for verifying an identity. When we prove an identity, we pick one side to work on and make substitutions until that side is transformed into the other side.

Example 7

Verifying the Identity Using Double-Angle Formulas and Reciprocal Identities

Verify the identity csc 2 θ2= cos(2θ) sin 2 θ . csc 2 θ2= cos(2θ) sin 2 θ .

Solution

For verifying this equation, we are bringing together several of the identities. We will use the double-angle formula and the reciprocal identities. We will work with the right side of the equation and rewrite it until it matches the left side.

cos(2θ) sin 2 θ = 12 sin 2 θ sin 2 θ = 1 sin 2 θ 2 sin 2 θ sin 2 θ = csc 2 θ2 cos(2θ) sin 2 θ = 12 sin 2 θ sin 2 θ = 1 sin 2 θ 2 sin 2 θ sin 2 θ = csc 2 θ2

Try It #5

Verify the identity tanθcotθ cos 2 θ= sin 2 θ. tanθcotθ cos 2 θ= sin 2 θ.

Media

Access these online resources for additional instruction and practice with the product-to-sum and sum-to-product identities.

  • Sum to Product Identities
  • Sum to Product and Product to Sum Identities

7.4 Section Exercises

Verbal

1.

Starting with the product to sum formula sinαcosβ= 1 2 [sin(α+β)+sin(αβ)], sinαcosβ= 1 2 [sin(α+β)+sin(αβ)], explain how to determine the formula for cosαsinβ. cosαsinβ.

2.

Explain two different methods of calculating cos( 195° )cos( 105° ), cos( 195° )cos( 105° ), one of which uses the product to sum. Which method is easier?

3.

Explain a situation where we would convert an equation from a sum to a product and give an example.

4.

Explain a situation where we would convert an equation from a product to a sum, and give an example.

Algebraic

For the following exercises, rewrite the product as a sum or difference.

5.

16sin(16x)sin(11x) 16sin(16x)sin(11x)

6.

20cos( 36t )cos( 6t ) 20cos( 36t )cos( 6t )

7.

2sin( 5x )cos( 3x ) 2sin( 5x )cos( 3x )

8.

10cos( 5x )sin( 10x ) 10cos( 5x )sin( 10x )

9.

sin( x )sin( 5x ) sin( x )sin( 5x )

10.

sin( 3x )cos( 5x ) sin( 3x )cos( 5x )

For the following exercises, rewrite the sum or difference as a product.

11.

cos( 6t )+cos( 4t ) cos( 6t )+cos( 4t )

12.

sin( 3x )+sin( 7x ) sin( 3x )+sin( 7x )

13.

cos( 7x )+cos( 7x ) cos( 7x )+cos( 7x )

14.

sin( 3x )sin( 3x ) sin( 3x )sin( 3x )

15.

cos( 3x )+cos( 9x ) cos( 3x )+cos( 9x )

16.

sinhsin( 3h ) sinhsin( 3h )

For the following exercises, evaluate the product for the following using a sum or difference of two functions.

17.

cos( 45° )cos( 15° ) cos( 45° )cos( 15° )

18.

cos( 45° )sin( 15° ) cos( 45° )sin( 15° )

19.

sin( −345° )sin( −15° ) sin( −345° )sin( −15° )

20.

sin( 195° )cos( 15° ) sin( 195° )cos( 15° )

21.

sin( −45° )sin( −15° ) sin( −45° )sin( −15° )

For the following exercises, evaluate the product using a sum or difference of two functions. Leave in terms of sine and cosine.

22.

cos( 23° )sin( 17° ) cos( 23° )sin( 17° )

23.

2sin( 100° )sin( 20° ) 2sin( 100° )sin( 20° )

24.

2sin(−100°)sin(−20°) 2sin(−100°)sin(−20°)

25.

sin( 213° )cos( ) sin( 213° )cos( )

26.

2cos(56°)cos(47°) 2cos(56°)cos(47°)

For the following exercises, rewrite the sum as a product of two functions. Leave in terms of sine and cosine.

27.

sin(76°)+sin(14°) sin(76°)+sin(14°)

28.

cos( 58° )cos( 12° ) cos( 58° )cos( 12° )

29.

sin(101°)sin(32°) sin(101°)sin(32°)

30.

cos( 100° )+cos( 200° ) cos( 100° )+cos( 200° )

31.

sin(−1°)+sin(−2°) sin(−1°)+sin(−2°)

For the following exercises, prove the identity.

32.

cos(a+b) cos(ab) = 1tanatanb 1+tanatanb cos(a+b) cos(ab) = 1tanatanb 1+tanatanb

33.

4sin( 3x )cos( 4x )=2sin( 7x )2sinx 4sin( 3x )cos( 4x )=2sin( 7x )2sinx

34.

6cos( 8x )sin( 2x ) sin( 6x ) =−3sin( 10x )csc( 6x )+3 6cos( 8x )sin( 2x ) sin( 6x ) =−3sin( 10x )csc( 6x )+3

35.

sinx+sin( 3x )=4sinx cos 2 x sinx+sin( 3x )=4sinx cos 2 x

36.

2( cos 3 xcosx sin 2 x )=cos( 3x )+cosx 2( cos 3 xcosx sin 2 x )=cos( 3x )+cosx

37.

2tanxcos( 3x )=secx( sin( 4x )sin( 2x ) ) 2tanxcos( 3x )=secx( sin( 4x )sin( 2x ) )

38.

cos( a+b )+cos( ab )=2cosacosb cos( a+b )+cos( ab )=2cosacosb

Numeric

For the following exercises, rewrite the sum as a product of two functions or the product as a sum of two functions. Give your answer in terms of sines and cosines. Then evaluate the final answer numerically, rounded to four decimal places.

39.

cos( 58 )+cos( 12 ) cos( 58 )+cos( 12 )

40.

sin( 2 )sin( 3 ) sin( 2 )sin( 3 )

41.

cos( 44 )cos( 22 ) cos( 44 )cos( 22 )

42.

cos( 176 )sin( 9 ) cos( 176 )sin( 9 )

43.

sin( 14 )sin( 85 ) sin( 14 )sin( 85 )

Technology

For the following exercises, algebraically determine whether each of the given expressions is a true identity. If it is not an identity, replace the right-hand side with an expression equivalent to the left side. Verify the results by graphing both expressions on a calculator.

44.

2sin(2x)sin(3x)=cosxcos(5x) 2sin(2x)sin(3x)=cosxcos(5x)

45.

cos( 10θ )+cos( 6θ ) cos( 6θ )cos( 10θ ) =cot( 2θ )cot( 8θ ) cos( 10θ )+cos( 6θ ) cos( 6θ )cos( 10θ ) =cot( 2θ )cot( 8θ )

46.

sin( 3x )sin( 5x ) cos( 3x )+cos( 5x ) =tanx sin( 3x )sin( 5x ) cos( 3x )+cos( 5x ) =tanx

47.

2cos(2x)cosx+sin(2x)sinx=2sinx 2cos(2x)cosx+sin(2x)sinx=2sinx

48.

sin( 2x )+sin( 4x ) sin( 2x )sin( 4x ) =tan( 3x )cotx sin( 2x )+sin( 4x ) sin( 2x )sin( 4x ) =tan( 3x )cotx

For the following exercises, simplify the expression to one term, then graph the original function and your simplified version to verify they are identical.

49.

sin( 9t )sin( 3t ) cos( 9t )+cos( 3t ) sin( 9t )sin( 3t ) cos( 9t )+cos( 3t )

50.

2sin( 8x )cos( 6x )sin( 2x ) 2sin( 8x )cos( 6x )sin( 2x )

51.

sin( 3x )sinx sinx sin( 3x )sinx sinx

52.

cos( 5x )+cos( 3x ) sin( 5x )+sin( 3x ) cos( 5x )+cos( 3x ) sin( 5x )+sin( 3x )

53.

sinxcos( 15x )cosxsin( 15x ) sinxcos( 15x )cosxsin( 15x )

Extensions

For the following exercises, prove the following sum-to-product formulas.

54.

sinxsiny=2sin( xy 2 )cos( x+y 2 ) sinxsiny=2sin( xy 2 )cos( x+y 2 )

55.

cosx+cosy=2cos( x+y 2 )cos( xy 2 ) cosx+cosy=2cos( x+y 2 )cos( xy 2 )

For the following exercises, prove the identity.

56.

sin(6x)+sin(4x) sin(6x)sin(4x) =tan(5x)cotx sin(6x)+sin(4x) sin(6x)sin(4x) =tan(5x)cotx

57.

cos(3x)+cosx cos(3x)cosx =cot(2x)cotx cos(3x)+cosx cos(3x)cosx =cot(2x)cotx

58.

cos(6y)+cos(8y) sin(6y)sin(4y) =cotycos(7y)sec(5y) cos(6y)+cos(8y) sin(6y)sin(4y) =cotycos(7y)sec(5y)

59.

cos( 2y )cos( 4y ) sin( 2y )+sin( 4y ) =tany cos( 2y )cos( 4y ) sin( 2y )+sin( 4y ) =tany

60.

sin( 10x )sin( 2x ) cos( 10x )+cos( 2x ) =tan( 4x ) sin( 10x )sin( 2x ) cos( 10x )+cos( 2x ) =tan( 4x )

61.

cosxcos(3x)=4 sin 2 xcosx cosxcos(3x)=4 sin 2 xcosx

62.

(cos(2x)cos(4x)) 2 + (sin(4x)+sin(2x)) 2 =4 sin 2 (3x) (cos(2x)cos(4x)) 2 + (sin(4x)+sin(2x)) 2 =4 sin 2 (3x)

63.

tan( π 4 t )= 1tant 1+tant tan( π 4 t )= 1tant 1+tant

7.4 Sum-to-Product and Product-to-Sum Formulas - Precalculus 2e | OpenStax (2024)

FAQs

How to solve a sum to product formula? ›

We can derive the sum to product formula using the product to sum formulas in trigonometry. We can apply these formulas to simplify trigonometric problems. The sum to product formula are: sin A + sin B = 2 sin [(A + B)/2] cos [(A - B)/2]

What is the formula for sum and product? ›

Sum to product formulas is used to find expression for sum and difference of sines and cosines functions as products of sine and cosine functions. Sum to product formulas in trigonometry are: sin A + sin B = 2 sin [(A + B)/2] cos [(A – B)/2] sin A – sin B = 2 sin [(A – B)/2] cos [(A + B)/2]

Where do the product to sum and sum to product formulas come from? ›

From the sum and difference identities, we can derive the product-to-sum formulas and the sum-to-product formulas for sine and cosine. We can use the product-to-sum formulas to rewrite products of sines, products of cosines, and products of sine and cosine as sums or differences of sines and cosines.

How many product to sum identities are there? ›

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) ]

How to memorize sum and difference formulas? ›

We can memorize all six sum and difference formulas by remembering only the sum formulas of the trigonometric functions. For difference formulas, we just need to interchange the signs '+' and '-'.

How to do the sum and product method? ›

  1. Find two numbers m and n whose product equals the value of a multiplied by c , and whose sum equals the value of b . Product =ac Sum =b.
  2. Decompose the term bx in the trinomial using the two numbers found.
  3. Factor by grouping.

How do you write a SUMPRODUCT formula? ›

To create the formula using our sample list above, type =SUMPRODUCT(C2:C5,D2:D5) and press Enter. Each cell in column C is multiplied by its corresponding cell in the same row in column D, and the results are added up.

What is sum of product AND product of sum? ›

The sum-of-products (SoP), which is also known as the disjunctive normal form. The product-of-sums (PoS), which is also known as the conjunctive normal form.

What is the formula for product? ›

In mathematics, a product is a number or a quantity obtained by multiplying two or more numbers together. For example: 4 × 7 = 28 Here, the number 28 is called the product of 4 and 7. As another example, the product of 6 and 4 is 24, because 6 times 4 is 24.

Is product the sum in math? ›

The outcome of adding two or more numbers gives the sum. The outcome of subtracting the two numbers gives the difference. The outcome of multiplying the two or more numbers gives the product. The result of the division of one number by another is the quotient.

What is the sum rule for sin? ›

Addition Formulas:

Sin(a+b)=sin(a)cos(b)+cos(a)sin(b). Cos(a+b)=cos(a)cos(b) - sin(a)sin(b). Tan(a+b)=(tan(a)+tan(b))/(1-tan(a)tan(b)).

What is the sin of 3x? ›

Sin3x is used to determine the value of the sine function for an angle that is thrice the measure of the angle x. The formula for the trigonometric function sin3x is given by, sin3x = 3 sin x - 4 sin3x.

How to prove product to sum formula? ›

The product to sum identities can be derived from the sum and difference identities. To derive the first identity, add the sine sum identity and the sine difference identity. To derive the second identity, subtract the sine difference identity from the sine sum identity.

What are the 48 formulas of trigonometry? ›

List of Trigonometry Formulas
  • sin²θ + cos²θ = 1.
  • tan2θ + 1 = sec2θ
  • cot2θ + 1 = cosec2θ
  • sin 2θ = 2 sin θ cos θ
  • cos 2θ = cos²θ – sin²θ
  • tan 2θ = 2 tan θ / (1 – tan²θ)
  • cot 2θ = (cot²θ – 1) / 2 cot θ

How many formulas are there in identities? ›

The Four Basic Identities in Maths Class 8 have Been Listed Below
Identity I(a+b)2 = a2+2ab+b2
Identity II(a-b)2 = a2- 2ab+b2
Identity IIIa2-b2= (a+b) (a-b)
Identity IV(x+a) (x+b) = x2+(a+b) x+ab

How do you calculate the sum of products? ›

You can calculate the sum of products in 3 steps:
  1. Determine the two series of numbers.
  2. Multiply the corresponding number, for example, a₁ × b₁.
  3. Add up results of the multiplications:
Jan 18, 2024

How do you calculate product formula? ›

For example, if cells A1 and A2 contain numbers, you can use the formula =PRODUCT(A1, A2) to multiply those two numbers together. You can also perform the same operation by using the multiply (*) mathematical operator; for example, =A1 * A2. The PRODUCT function is useful when you need to multiply many cells together.

What is the sum rule for products? ›

The rule of sum says that if there are n choices for one task and m choices for another task, then there are n + m n+m n+m ways to do one of these tasks if these two tasks cannot be done at the same time.

What is the rule of sum and product in problem solving? ›

If two events are not mutually exclusive (that is, we do them separately), then we apply the product rule. n1 · n2 ways of doing the overall procedure. If two events are mutually exclusive, that is, they cannot be done at the same time, then we must apply the sum rule.

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