Sum and Difference Identities for Cosine — Transcript

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1. 0:05Welcome to a video on the sum and  difference identities for cosine.
2. 0:09The goals of the video are to use these  identities to determine function values.
3. 0:14If we have the cosine of a sum or  difference, here are the identities:
4. 0:18The cosine of the quantity A plus B is equal to  cosine A times cosine B minus sine A times sine B.
5. 0:28If we have a difference, it's equal to cosine  A times cosine B plus sine A times sine B.
6. 0:36Instead of writing this as two different  identities, sometimes it's written like this.
7. 0:39Notice here we have a plus or minus sign  and over here we have a minus plus sign.
8. 0:44If we use the addition sign here,  we would use the subtraction sign,
9. 0:48and if we use the subtraction sign  here, we would use the addition sign.
10. 0:55Now, I am including a slide that  shows the proof of these identities.
11. 0:59This proof is pretty straightforward if you set
12. 1:01up this diagram using the instructions  and then follow the proof on the left.
13. 1:07You may want to pause the video and take a look  at this to see where these identities come from.
14. 1:13Let's take a look at a couple of  problems using the identities.
15. 1:17Here's a problem where we have sine A  as 12 over 13 in the second quadrant,
16. 1:23and sine B as 4 over 5 in the first quadrant.
17. 1:27We want to find the cosine  of the quantity A plus B.
18. 1:31Let's go ahead and write out our identity.
19. 1:33Notice if we have a sum, then we're going  to use a difference of these two products.
20. 1:39We are given sine A and sine B, but we  still have to find cosine A and cosine B.
21. 1:46Let's go ahead and sketch these  angles in standard position.
22. 1:49For angle A in the second quadrant,  it might look something like this.
23. 1:55Our reference angle, or A  prime, would be this angle here.
24. 1:59The opposite and adjacent sides would be 12  over 13, and this is a 5 12 13 right triangle.
25. 2:11For angle B in the first quadrant,  it has a sine of 4 over 5.
26. 2:16If we call the angle B, this would be the  opposite side and this would be the hypotenuse.
27. 2:21This is a 3 4 5 right triangle, so  this side here would have length 3.
28. 2:27We use this information to complete our identity.
29. 2:32The cosine of angle A would be the  cosine of this reference angle,
30. 2:36which is adjacent over hypotenuse or -5 over 13.
31. 2:44Times the cosine of angle B, which would  be adjacent over hypotenuse or 3 over 5.
32. 2:54Minus sine A times sine B.
33. 2:57Sine A is 12 over 13 and sine B is 4 over 5.
34. 3:03This product would be -5 over 65 plus  -48 over 65, which gives us -63 over 65.
35. 3:17For the cosine of angle A plus B,  this is what we were trying to find.
36. 3:22So let's go ahead and state it one more time:  cosine of A plus B is equal to -63 over 65.
37. 3:30Let's go ahead and take a look  at a couple more problems.
38. 3:33Here we want to determine the  exact value of cosine 15 degrees.
39. 3:3715 degrees is not one of  those nice reference angles,
40. 3:41but we can use a sum or difference of two angles.
41. 3:49For example, if we use the reference angle of
42. 3:5245 degrees and then subtract the  reference angle of 30 degrees.
43. 3:56That would give us an angle of 15 degrees.
44. 3:59So we can use angle A equal to 45  degrees and angle B equal to 30 degrees.
45. 4:05Let's set this up: cosine of 45 degrees minus  30 degrees is equal to cosine A times cosine B.
46. 4:20Now we use the subtraction sign  here, so we'll use an addition sign,
47. 4:25and then it’s going to be sine A times sine B.
48. 4:33Now we’re going to fill in these values,  find these products, and then find the sum.
49. 4:38We’ll replace this now with 15 degrees.
50. 4:44The cosine of 45 degrees is equal to  the x coordinate on the unit circle,
51. 4:49so we have square root 2 over 2.
52. 4:52Times the cosine of 30 degrees, so our  x coordinate is square root 3 over 2.
53. 5:00Plus sine of 45 degrees is also square root  2 over 2, and sine of 30 degrees is 1/2.
54. 5:11So we have square root 6 over  4 plus square root 2 over 4.
55. 5:17So now we know the cosine of 15 degrees is
56. 5:21exactly equal to the sum of square  root 6 plus square root 2 over 4.
57. 5:28Now we can check this on our  graphing calculators in decimal form.
58. 5:31Let’s go ahead and do that first.
59. 5:33Let’s check to make sure we’re  in degree mode, press enter,
60. 5:38go back to the home screen, and  we’ll type in cosine 15 degrees.
61. 5:43Then we’ll compare this to what we found to be  the exact value, and this verifies our work.
62. 5:58Here’s another one, same  idea, but now it’s in radians.
63. 6:02Sometimes it’s difficult to determine  which reference angles we can add and
64. 6:05subtract to come up with this angle in radians.
65. 6:08Let’s convert this to degrees first.
66. 6:15This is equal to 105 degrees, so  that should tell us if we use a
67. 6:2060 degree angle and we add a 45 degree  angle, that would give us 105 degrees.
68. 6:29But let’s go ahead and be  consistent and use radians.
69. 6:3260 degrees is equal to pi over 3 radians and  45 degrees is equal to pi over 4 radians.
70. 6:42What this tells us is that cosine of 7 pi over 12
71. 6:44is equal to the cosine of  pi over 3 plus pi over 4.
72. 6:52These are the angles we’ll use in our identity.
73. 6:55Again, we have cosine A times cosine B.
74. 7:02Here we’re using a sum, which means  we use a difference in our identity,
75. 7:07and then sine A times sine B.
76. 7:12Cosine of pi over 3 radians,  so that’s equal to 1/2.
77. 7:21The cosine of pi over 4 radians  is square root 2 over 2.
78. 7:26The sine of pi over 3 radians  would be square root 3 over 2,
79. 7:32and the sine of pi over 4 radians  would be square root 2 over 2.
80. 7:39Notice we have a common denominator of 4.
81. 7:42So what we have found is that  cosine of 7 pi over 12 radians
82. 7:47is equal to the numerator of square  root 2 minus square root 6 over 4.
83. 8:00I think I have one more question.
84. 8:01If we have sine 40 degrees times cosine 50 degrees  plus sine 40 degrees times sine 50 degrees, what
85. 8:12we should notice here is that it fits our  identity when we're using a difference.
86. 8:17So we’re going to rewrite  this in this form this time.
87. 8:20If we’re using a subtraction sign here,
88. 8:23we’re going to find the sum of angle A and  angle B, which are 40 degrees and 50 degrees.
89. 8:31So this would be equal to the cosine of 40 degrees  plus 50 degrees, which is equal to 90 degrees.
90. 8:40Well, the cosine of 90 degrees,  here we are in the unit circle,
91. 8:43remember cosine is equal to  x, so this is equal to zero.
92. 8:49And that’s all there is to this problem.
93. 8:51Okay, I hope you found this video helpful.  Thank you for watching and have a nice day.