Trig products as sum and differecnes
Download as KEY, PDF0 likes3,819 views
The document presents trigonometric formulae for expressing the product of trigonometric functions as sums or differences. It shows that: 1) sin(A+B) + sin(A-B) = 2sinAcosB 2) sin(A+B) - sin(A-B) = 2cosAsinB 3) cos(A+B) + cos(A-B) = 2cosAcosB 4) cos(A-B) - cos(A+B) = 2sinAsinB It also derives equivalent formulae using the sums and differences of the arguments.
Trig products as sum and differecnes
- 1. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011 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
- 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
- 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
- 17. 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
- 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
- 28. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011 Example 3 Express cos 3x + cos 7x as a product
- 29. 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
- 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