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More from Simon Borgert (17)
Trig products as sum and differecnes
- 3. 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
2sin A cos B = sin(A + B) + sin(A − B)
or
- 4. 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
2sin A cos B = sin(A + B) + sin(A − B)
or
2sin A cos B = sin(sum) + sin(difference)
- 5. 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
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
- 6. 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
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)
- 10. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
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
- 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
- 18. 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)
- 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
- 22. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
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)
- 23. 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)
- 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)
- 25. 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)
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
- 27. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011
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
- 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
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