1. 5.3/5.4 – Sum and Difference Identities
Math 150
5.3/5.4 – Sum and Difference Identities

2. Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵. Take 𝐴= 𝜋 2 and 𝐵=0 for example
Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵 . Take 𝐴= 𝜋 2 and 𝐵=0 for example. cos 𝐴−𝐵 = cos 𝜋 2 −0 = cos 𝜋 2 =𝟎 …but… cos 𝐴 − cos 𝐵 = cos 𝜋 2 − cos 0 =0−1=−𝟏

3. Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵. Take 𝐴= 𝜋 2 and 𝐵=0 for example
Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵 . Take 𝐴= 𝜋 2 and 𝐵=0 for example. cos 𝐴−𝐵 = cos 𝜋 2 −0 = cos 𝜋 2 =𝟎 …but… cos 𝐴 − cos 𝐵 = cos 𝜋 2 − cos 0 =0−1=−𝟏

4. Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵. Take 𝐴= 𝜋 2 and 𝐵=0 for example
Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵 . Take 𝐴= 𝜋 2 and 𝐵=0 for example. cos 𝐴−𝐵 = cos 𝜋 2 −0 = cos 𝜋 2 =𝟎 …but… cos 𝐴 − cos 𝐵 = cos 𝜋 2 − cos 0 =0−1=−𝟏

5. Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵. Take 𝐴= 𝜋 2 and 𝐵=0 for example
Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵 . Take 𝐴= 𝜋 2 and 𝐵=0 for example. cos 𝐴−𝐵 = cos 𝜋 2 −0 = cos 𝜋 2 =𝟎 …but… cos 𝐴 − cos 𝐵 = cos 𝜋 2 − cos 0 =0−1=−𝟏

6. Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵. Take 𝐴= 𝜋 2 and 𝐵=0 for example
Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵 . Take 𝐴= 𝜋 2 and 𝐵=0 for example. cos 𝐴−𝐵 = cos 𝜋 2 −0 = cos 𝜋 2 =𝟎 …but… cos 𝐴 − cos 𝐵 = cos 𝜋 2 − cos 0 =0−1=−𝟏

7. Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵. Take 𝐴= 𝜋 2 and 𝐵=0 for example
Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵 . Take 𝐴= 𝜋 2 and 𝐵=0 for example. cos 𝐴−𝐵 = cos 𝜋 2 −0 = cos 𝜋 2 =𝟎 …but… cos 𝐴 − cos 𝐵 = cos 𝜋 2 − cos 0 =0−1=−𝟏

8. Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵. Take 𝐴= 𝜋 2 and 𝐵=0 for example
Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵 . Take 𝐴= 𝜋 2 and 𝐵=0 for example. cos 𝐴−𝐵 = cos 𝜋 2 −0 = cos 𝜋 2 =𝟎 …but… cos 𝐴 − cos 𝐵 = cos 𝜋 2 − cos 0 =0−1=−𝟏

9. Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵. Take 𝐴= 𝜋 2 and 𝐵=0 for example
Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵 . Take 𝐴= 𝜋 2 and 𝐵=0 for example. cos 𝐴−𝐵 = cos 𝜋 2 −0 = cos 𝜋 2 =𝟎 …but… cos 𝐴 − cos 𝐵 = cos 𝜋 2 − cos 0 =0−1=−𝟏

10. Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵. Take 𝐴= 𝜋 2 and 𝐵=0 for example
Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵 . Take 𝐴= 𝜋 2 and 𝐵=0 for example. cos 𝐴−𝐵 = cos 𝜋 2 −0 = cos 𝜋 2 =𝟎 …but… cos 𝐴 − cos 𝐵 = cos 𝜋 2 − cos 0 =0−1=−𝟏

11. Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵. Take 𝐴= 𝜋 2 and 𝐵=0 for example
Note that cos 𝐴−𝐵 ≠ cos 𝐴 − cos 𝐵 . Take 𝐴= 𝜋 2 and 𝐵=0 for example. cos 𝐴−𝐵 = cos 𝜋 2 −0 = cos 𝜋 2 =𝟎 …but… cos 𝐴 − cos 𝐵 = cos 𝜋 2 − cos 0 =0−1=−𝟏

12. So what is cos (𝐴−𝐵) equal to?

19. cos 𝐴 − cos 𝐵 sin 𝐴 − sin 𝐵 2

20. cos 𝐴 − cos 𝐵 sin 𝐴 − sin 𝐵 2

21. cos 𝐴 − cos 𝐵 sin 𝐴 − sin 𝐵 2
cos 𝐴−𝐵 − sin 𝐴−𝐵 −0 2

22. Using the distance formula twice, we get: cos 𝐴−𝐵 −1 2 + sin 𝐴−𝐵 −0 2 = cos 𝐴 − cos 𝐵 2 + sin 𝐴 − sin 𝐵 2 cos 2 (𝐴−𝐵) −2 cos 𝐴−𝐵 +1+ sin 2 𝐴−𝐵 = cos 2 𝐴 −2 cos 𝐴 cos 𝐵 + cos 2 𝐵 + sin 2 𝐴 −2 sin 𝐴 sin 𝐵 + sin 2 𝐵 2−2 cos 𝐴−𝐵 =2−2 cos 𝐴 cos 𝐵 −2 sin 𝐴 sin 𝐵 −2 cos 𝐴−𝐵 =−2 cos 𝐴 cos 𝐵 −2 sin 𝐴 sin 𝐵 𝐜𝐨𝐬 𝑨−𝑩 = 𝐜𝐨𝐬 𝑨 𝐜𝐨𝐬 𝑩 + 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩

23. Using the distance formula twice, we get: cos 𝐴−𝐵 −1 2 + sin 𝐴−𝐵 −0 2 = cos 𝐴 − cos 𝐵 2 + sin 𝐴 − sin 𝐵 2 cos 2 (𝐴−𝐵) −2 cos 𝐴−𝐵 +1+ sin 2 𝐴−𝐵 = cos 2 𝐴 −2 cos 𝐴 cos 𝐵 + cos 2 𝐵 + sin 2 𝐴 −2 sin 𝐴 sin 𝐵 + sin 2 𝐵 2−2 cos 𝐴−𝐵 =2−2 cos 𝐴 cos 𝐵 −2 sin 𝐴 sin 𝐵 −2 cos 𝐴−𝐵 =−2 cos 𝐴 cos 𝐵 −2 sin 𝐴 sin 𝐵 𝐜𝐨𝐬 𝑨−𝑩 = 𝐜𝐨𝐬 𝑨 𝐜𝐨𝐬 𝑩 + 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩

24. Using the distance formula twice, we get: cos 𝐴−𝐵 −1 2 + sin 𝐴−𝐵 −0 2 = cos 𝐴 − cos 𝐵 2 + sin 𝐴 − sin 𝐵 2 cos 2 (𝐴−𝐵) −2 cos 𝐴−𝐵 +1+ sin 2 𝐴−𝐵 = cos 2 𝐴 −2 cos 𝐴 cos 𝐵 + cos 2 𝐵 + sin 2 𝐴 −2 sin 𝐴 sin 𝐵 + sin 2 𝐵 2−2 cos 𝐴−𝐵 =2−2 cos 𝐴 cos 𝐵 −2 sin 𝐴 sin 𝐵 −2 cos 𝐴−𝐵 =−2 cos 𝐴 cos 𝐵 −2 sin 𝐴 sin 𝐵 𝐜𝐨𝐬 𝑨−𝑩 = 𝐜𝐨𝐬 𝑨 𝐜𝐨𝐬 𝑩 + 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩

25. Using the distance formula twice, we get: cos 𝐴−𝐵 −1 2 + sin 𝐴−𝐵 −0 2 = cos 𝐴 − cos 𝐵 2 + sin 𝐴 − sin 𝐵 2 cos 2 (𝐴−𝐵) −2 cos 𝐴−𝐵 +1+ sin 2 𝐴−𝐵 = cos 2 𝐴 −2 cos 𝐴 cos 𝐵 + cos 2 𝐵 + sin 2 𝐴 −2 sin 𝐴 sin 𝐵 + sin 2 𝐵 2−2 cos 𝐴−𝐵 =2−2 cos 𝐴 cos 𝐵 −2 sin 𝐴 sin 𝐵 −2 cos 𝐴−𝐵 =−2 cos 𝐴 cos 𝐵 −2 sin 𝐴 sin 𝐵 𝐜𝐨𝐬 𝑨−𝑩 = 𝐜𝐨𝐬 𝑨 𝐜𝐨𝐬 𝑩 + 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩

26. Using the distance formula twice, we get: cos 𝐴−𝐵 −1 2 + sin 𝐴−𝐵 −0 2 = cos 𝐴 − cos 𝐵 2 + sin 𝐴 − sin 𝐵 2 cos 2 (𝐴−𝐵) −2 cos 𝐴−𝐵 +1+ sin 2 𝐴−𝐵 = cos 2 𝐴 −2 cos 𝐴 cos 𝐵 + cos 2 𝐵 + sin 2 𝐴 −2 sin 𝐴 sin 𝐵 + sin 2 𝐵 2−2 cos 𝐴−𝐵 =2−2 cos 𝐴 cos 𝐵 −2 sin 𝐴 sin 𝐵 −2 cos 𝐴−𝐵 =−2 cos 𝐴 cos 𝐵 −2 sin 𝐴 sin 𝐵 𝐜𝐨𝐬 𝑨−𝑩 = 𝐜𝐨𝐬 𝑨 𝐜𝐨𝐬 𝑩 + 𝐬𝐢𝐧 𝑨 𝐬𝐢𝐧 𝑩

27. To find cos (𝐴+𝐵) , we can rewrite it as cos (𝐴− −𝐵 ) : cos 𝐴+𝐵 = cos 𝐴− −𝐵 = cos 𝐴 cos −𝐵 + sin 𝐴 sin −𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵

28. To find cos (𝐴+𝐵) , we can rewrite it as cos (𝐴− −𝐵 ) : cos 𝐴+𝐵 = cos 𝐴− −𝐵 = cos 𝐴 cos −𝐵 + sin 𝐴 sin −𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵

29. To find cos (𝐴+𝐵) , we can rewrite it as cos (𝐴− −𝐵 ) : cos 𝐴+𝐵 = cos 𝐴− −𝐵 = cos 𝐴 cos −𝐵 + sin 𝐴 sin −𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵

30. To find cos (𝐴+𝐵) , we can rewrite it as cos (𝐴− −𝐵 ) : cos 𝐴+𝐵 = cos 𝐴− −𝐵 = cos 𝐴 cos −𝐵 + sin 𝐴 sin −𝐵 = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵

31. To find sin (𝐴+𝐵) , we can make use of sin 𝜃 = cos ( 90 ∘ −𝜃):
= cos 90 ∘ −𝐴 cos 𝐵 + sin 90 ∘ −𝐴 sin 𝐵
= sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵

32. To find sin (𝐴+𝐵) , we can make use of sin 𝜃 = cos ( 90 ∘ −𝜃):
= cos 90 ∘ −𝐴 cos 𝐵 + sin 90 ∘ −𝐴 sin 𝐵
= sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵

33. To find sin (𝐴+𝐵) , we can make use of sin 𝜃 = cos ( 90 ∘ −𝜃):
= cos 90 ∘ −𝐴 cos 𝐵 + sin 90 ∘ −𝐴 sin 𝐵
= sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵

34. To find sin (𝐴+𝐵) , we can make use of sin 𝜃 = cos ( 90 ∘ −𝜃):
= cos 90 ∘ −𝐴 cos 𝐵 + sin 90 ∘ −𝐴 sin 𝐵
= sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵

35. To find sin (𝐴+𝐵) , we can make use of sin 𝜃 = cos ( 90 ∘ −𝜃):
= cos 90 ∘ −𝐴 cos 𝐵 + sin 90 ∘ −𝐴 sin 𝐵
= sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵

36. To find sin 𝐴−𝐵 , we can rewrite it as sin 𝐴+ −𝐵 : sin 𝐴−𝐵 = sin 𝐴+ −𝐵 = sin 𝐴 cos −𝐵 + cos 𝐴 sin −𝐵 = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵

37. To find sin 𝐴−𝐵 , we can rewrite it as sin 𝐴+ −𝐵 : sin 𝐴−𝐵 = sin 𝐴+ −𝐵 = sin 𝐴 cos −𝐵 + cos 𝐴 sin −𝐵 = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵

38. To find sin 𝐴−𝐵 , we can rewrite it as sin 𝐴+ −𝐵 : sin 𝐴−𝐵 = sin 𝐴+ −𝐵 = sin 𝐴 cos −𝐵 + cos 𝐴 sin −𝐵 = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵

39. To find sin 𝐴−𝐵 , we can rewrite it as sin 𝐴+ −𝐵 : sin 𝐴−𝐵 = sin 𝐴+ −𝐵 = sin 𝐴 cos −𝐵 + cos 𝐴 sin −𝐵 = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵

40. Sum/difference identities for tangent can be derived using the sum/difference identities for sine and cosine. So, in summary…

41. Sum/difference identities for tangent can be derived using the sum/difference identities for sine and cosine. So, in summary…

42. Sum/Difference Identities sin 𝐴+𝐵 = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵 sin (𝐴−𝐵) = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵 cos 𝐴+𝐵 = cos 𝐴 cos 𝐵 − sin 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 tan 𝐴−𝐵 = tan 𝐴 − tan 𝐵 1+ tan 𝐴 tan 𝐵

43. Ex 1. Use identities to find each exact value
Ex 1. Use identities to find each exact value. cos 15 ∘ sin 5𝜋 12 tan 7𝜋 12

44. Ex 2. Find cos 𝑠+𝑡 and sin 𝑠−𝑡 if sin 𝑠 = 3 5 and cos 𝑡 =− and both 𝑠 and 𝑡 are in QII. Also, which quadrant is 𝑠+𝑡 in?

45. Ex 3. Verify the following identity
Ex 3. Verify the following identity. sin 𝑥+𝑦 cos 𝑥 cos 𝑦 = tan 𝑥 + tan 𝑦