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Trig products as sum and differecnes

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Trig products as sum and differecnes

  1. 1. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011 Further Trig Formulae
  2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 28. http://simonborgert.com http://onlinemaths.net © Simon Borgert 2011 Example 3 Express cos 3x + cos 7x as a product
  29. 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. 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. 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. 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. 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. 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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