Presentation that explains how the product to sum and sum to product formula's are developed.

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Further Trig Formulae

2. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Products as Sums or
Differences
sin(A + B) = sin A cos B + cos Asin B
sin(A − B) = sin A cos B − cos Asin B
Add the two together

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Products as Sums or
Differences
sin(A + B) = sin A cos B + cos Asin B
sin(A − B) = sin A cos B − cos Asin B
Add the two together
2sin A cos B = sin(A + B) + sin(A − B)
or

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Products as Sums or
Differences
sin(A + B) = sin A cos B + cos Asin B
sin(A − B) = sin A cos B − cos Asin B
Add the two together
2sin A cos B = sin(A + B) + sin(A − B)
or
2sin A cos B = sin(sum) + sin(difference)

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Products as Sums or
Differences
sin(A + B) = sin A cos B + cos Asin B
sin(A − B) = sin A cos B − cos Asin B
Add the two together
2sin A cos B = sin(A + B) + sin(A − B)
or
2sin A cos B = sin(sum) + sin(difference)
1 1
sin A cos B = sin(sum) + sin(difference)
2 2

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Products as Sums or
Differences
sin(A + B) = sin A cos B + cos Asin B
sin(A − B) = sin A cos B − cos Asin B
Subtract the second from the first
2 cos Asin B = sin(A + B) − sin(A − B)
or
2 cos Asin B = sin(sum) − sin(difference)
1 1
cos Asin B = sin(sum) − sin(difference)
2 2

7. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Products as Sums or
Differences
cos(A + B) = cos A cos B − sin Asin B
cos(A − B) = cos A cos B + sin Asin B
Add the two together
2 cos A cos B = cos(A + B) + cos(A − B)
or
2 cos A cos B = cos(sum) + cos(difference)
1 1
cos A cos B = cos(sum) + cos(difference)
2 2

8. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Products as Sums or
Differences
cos(A + B) = cos A cos B − sin Asin B
cos(A − B) = cos A cos B + sin Asin B
Subtract the first from the second
2sin Asin B = cos(A − B) − cos(A + B)
or
2sin Asin B = cos(difference) − cos(sum)
1 1
sin Asin B = cos(difference) − cos(sum)
2 2

9. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Summary
2sin A cos B = sin(sum) + sin(difference)
2 cos Asin B = sin(sum) − sin(difference)
2 cos A cos B = cos(sum) + cos(difference)
2sin Asin B = cos(difference) − cos(sum)

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or...
1 1
sin A cos B = sin(sum) + sin(difference)
2 2
1 1
cos Asin B = sin(sum) − sin(difference)
2 2
1 1
cos A cos B = cos(sum) + cos(difference)
2 2
1 1
sin Asin B = cos(difference) − cos(sum)
2 2

11. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 1
Express the product of cos 5x sin 3x as a sum of trigonometric functions

12. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 1
Express the product of cos 5x sin 3x as a sum of trigonometric functions
Step 1: Identify the product to sum formula

13. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 1
Express the product of cos 5x sin 3x as a sum of trigonometric functions
Step 1: Identify the product to sum formula
1 1
cos Asin B = sin(sum) − sin(difference)
2 2

14. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 1
Express the product of cos 5x sin 3x as a sum of trigonometric functions
Step 1: Identify the product to sum formula
1 1
cos Asin B = sin(sum) − sin(difference)
2 2
Step 2: Apply it to the given product

15. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 1
Express the product of cos 5x sin 3x as a sum of trigonometric functions
Step 1: Identify the product to sum formula
1 1
cos Asin B = sin(sum) − sin(difference)
2 2
Step 2: Apply it to the given product
1 1
cos 5x sin 3x = sin(5x + 3x) − sin(5x − 3x)
2 2

16. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 1
Express the product of cos 5x sin 3x as a sum of trigonometric functions
Step 1: Identify the product to sum formula
1 1
cos Asin B = sin(sum) − sin(difference)
2 2
Step 2: Apply it to the given product
1 1
cos 5x sin 3x = sin(5x + 3x) − sin(5x − 3x)
2 2
1 1
= sin(8x) − sin(2x)
2 2

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Example 2
Express the product of 2sin3x sin 2x as a difference of trigonometric functions
Step 1: Identify the product to sum formula

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Example 2
Express the product of 2sin3x sin 2x as a difference of trigonometric functions
Step 1: Identify the product to sum formula
2sin Asin B = cos(difference) − cos(sum)

19. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 2
Express the product of 2sin3x sin 2x as a difference of trigonometric functions
Step 1: Identify the product to sum formula
2sin Asin B = cos(difference) − cos(sum)
Step 2: Apply it to the given product

20. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 2
Express the product of 2sin3x sin 2x as a difference of trigonometric functions
Step 1: Identify the product to sum formula
2sin Asin B = cos(difference) − cos(sum)
Step 2: Apply it to the given product
2sin 3x sin 2x = cos(3x − 2x) − cos(3x + 2x)

21. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 2
Express the product of 2sin3x sin 2x as a difference of trigonometric functions
Step 1: Identify the product to sum formula
2sin Asin B = cos(difference) − cos(sum)
Step 2: Apply it to the given product
2sin 3x sin 2x = cos(3x − 2x) − cos(3x + 2x)
= cos x − cos 5x

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Sums and Differences as Products
2sin A cos B = sin(A + B) + sin(A − B)
2 cos Asin B = sin(A + B) − sin(A − B)
2 cos A cos B = cos(A + B) + cos(A − B)
2sin Asin B = cos(A − B) − cos(A + B)

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Sums and Differences as Products
Let U = A + B and V = A − B
2sin A cos B = sin(A + B) + sin(A − B)
U +V U −V
2 cos Asin B = sin(A + B) − sin(A − B) ∴A = and B =
2 cos A cos B = cos(A + B) + cos(A − B)
2 2
2sin Asin B = cos(A − B) − cos(A + B)

24. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Sums and Differences as Products
Let U = A + B and V = A − B
2sin A cos B = sin(A + B) + sin(A − B)
U +V U −V
2 cos Asin B = sin(A + B) − sin(A − B) ∴A = and B =
2 cos A cos B = cos(A + B) + cos(A − B)
2 2
2sin Asin B = cos(A − B) − cos(A + B)

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Sums and Differences as Products
Let U = A + B and V = A − B
2sin A cos B = sin(A + B) + sin(A − B)
U +V U −V
2 cos Asin B = sin(A + B) − sin(A − B) ∴A = and B =
2 cos A cos B = cos(A + B) + cos(A − B)
2 2
2sin Asin B = cos(A − B) − cos(A + B)
so the formulas become

26. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Sums and Differences as Products
Let U = A + B and V = A − B
2sin A cos B = sin(A + B) + sin(A − B)
U +V U −V
2 cos Asin B = sin(A + B) − sin(A − B) ∴A = and B =
2 cos A cos B = cos(A + B) + cos(A − B)
2 2
U +V  U −V 
sinU + sinV = 2sin   cos 
2sin Asin B = cos(A − B) − cos(A + B)  2   2  
U +V  U −V 
sinU − sinV = 2 cos   sin 
 2   2  
U +V  U −V 
cosU + cosV = 2 cos   cos 
so the formulas become  2   2  
U +V  U −V 
cosV − cosU = 2sin   sin 
 2   2  
U +V  U −V 
cosU − cosV = −2sin   sin 
 2   2  

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Sums and Differences as Products
U +V  U −V 
sinU + sinV = 2sin   cos   1  1 
 2   2  sin+ sin = 2sin  sum  cos  difference
2  2 
U +V  U −V 
sinU − sinV = 2 cos   sin 
 2   2   1  1 
sin− sin = 2 cos  sum  sin  difference
2  2 
U +V  U −V 
cosU + cosV = 2 cos   cos 
 2   2   1  1 
cos+ cos = 2 cos  sum  cos  difference
2  2 
U +V  U −V 
cosV − cosU = 2sin   sin   1  1 
 2   2  cos− cos = −2sin  sum  sin  difference
2  2 
U +V  U −V 
cosU − cosV = −2sin   sin  
 2   2 

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Example 3
Express cos 3x + cos 7x as a product

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Example 3
Express cos 3x + cos 7x as a product
Step 1: Identify the sum to product formula

30. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 3
Express cos 3x + cos 7x as a product
Step 1: Identify the sum to product formula
1  1 
cos+ cos = 2 cos  sum  cos  difference
2  2 

31. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 3
Express cos 3x + cos 7x as a product
Step 1: Identify the sum to product formula
1  1 
cos+ cos = 2 cos  sum  cos  difference
2  2 
Step 2: Apply it to the given product

32. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 3
Express cos 3x + cos 7x as a product
Step 1: Identify the sum to product formula
1  1 
cos+ cos = 2 cos  sum  cos  difference
2  2 
Step 2: Apply it to the given product
1  1 
cos 3x + cos 7x = 2 cos  (3x + 7x) cos  (3x − 7x)
2  2 

33. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 3
Express cos 3x + cos 7x as a product
Step 1: Identify the sum to product formula
1  1 
cos+ cos = 2 cos  sum  cos  difference
2  2 
Step 2: Apply it to the given product
1  1 
cos 3x + cos 7x = 2 cos  (3x + 7x) cos  (3x − 7x)
2  2 
= 2 cos ( 5x ) cos ( −4x )

34. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
Example 3
Express cos 3x + cos 7x as a product
Step 1: Identify the sum to product formula
1  1 
cos+ cos = 2 cos  sum  cos  difference
2  2 
Step 2: Apply it to the given product
1  1 
cos 3x + cos 7x = 2 cos  (3x + 7x) cos  (3x − 7x)
2  2 
= 2 cos ( 5x ) cos ( −4x )
= 2 cos ( 5x ) cos ( 4x ) as cos(−x) = cos x