Sum and Difference Identities for Sine — Transcript

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1. 0:11- WELCOME TO A VIDEO
2. 0:12ON THE SUM AND DIFFERENCE IDENTITIES FOR SINE.
3. 0:14THE GOAL OF THE VIDEO
4. 0:15IS TO USE THE SUM AND DIFFERENCE IDENTITIES
5. 0:17TO DETERMINE FUNCTION VALUES.
6. 0:20SO THE SINE OF A SUM OR DIFFERENCE
7. 0:22ARE EQUAL TO THE FOLLOWING.
8. 0:24THE SINE OF THE QUANTITY A + B
9. 0:25IS EQUAL TO SINE A x COSINE B + COSINE A x SINE B.
10. 0:32AND IF WE HAVE A DIFFERENCE,
11. 0:33OR THE SINE OF THE QUANTITY A - B,
12. 0:36THAT'S EQUAL TO SINE A x COSINE B - COSINE A x SINE B.
13. 0:42NOW, WE CAN COMBINE THESE TWO IDENTITIES
14. 0:44INTO A SINGLE IDENTITY IF WE WRITE IT LIKE THIS.
15. 0:47NOTICE WE HAVE A +/- SIGN HERE AND A +/- SIGN HERE.
16. 0:51SO IF WE HAVE A SUM OF TWO ANGLES,
17. 0:52WE HAVE A SUM OF THESE PRODUCTS.
18. 0:54IF WE HAVE A DIFFERENCE OF TWO ANGLES,
19. 0:56WE HAVE THE DIFFERENCE OF THESE TWO PRODUCTS.
20. 0:59NOW, I DID INCLUDE A SLIDE
21. 1:01THAT SHOWS THE PROOF OF THE SINE SUM IDENTITY.
22. 1:04WE DON'T HAVE TIME TO GO THROUGH ALL OF THIS NOW,
23. 1:07BUT IF YOU READ THROUGH THIS, IT'S PRETTY STRAIGHTFORWARD,
24. 1:10AND IT WILL VERIFY THE VALIDITY
25. 1:11OF THE SUM AND DIFFERENCE IDENTITY FOR SINE.
26. 1:16LET'S GO AHEAD AND TAKE A LOOK AT SOME PROBLEMS.
27. 1:19WE WANT TO DETERMINE THE SINE OF A - B
28. 1:22GIVEN SINE A EQUALS 4/5 IN THE SECOND QUADRANT,
29. 1:26AND COSINE B = -5/13 IN THE THIRD QUADRANT.
30. 1:31LET'S GO AHEAD AND SKETCH THESE ANGLES
31. 1:34IN STANDARD POSITION.
32. 1:36SO FOR ANGLE WHOSE TERMINAL SIDE
33. 1:38IS IN THE SECOND QUADRANT,
34. 1:41IT MAY LOOK SOMETHING LIKE THIS,
35. 1:45AND SO WE'LL CALL THIS REFERENCE ANGLE A PRIME.
36. 1:48SINCE THE SINE OF THIS ANGLE IS EQUAL TO 4/5,
37. 1:51WE KNOW THE OPPOSITE SIDE IS EQUAL TO 4,
38. 1:53THE HYPOTENUSE IS EQUAL TO 5.
39. 1:55THIS IS A 3, 4, 5 RIGHT TRIANGLE,
40. 1:57SO THIS WOULD BE EQUAL TO -3
41. 1:59SINCE WE ARE IN THE SECOND QUADRANT.
42. 2:02NEXT, COSINE B IS EQUAL TO -5/13 IN THE THIRD QUADRANT.
43. 2:10SO THIS WOULD BE ANGLE B IN STANDARD POSITION.
44. 2:13SO IF WE DRAW OUR REFERENCE TRIANGLE,
45. 2:16IT MIGHT LOOK SOMETHING LIKE THIS,
46. 2:18SO WE'LL CALL OUR REFERENCE ANGLE B PRIME
47. 2:21AND THE COSINE OF THIS ANGLE IS EQUAL TO -5/13,
48. 2:27AND THIS IS 5, 12, 13 RIGHT TRIANGLE,
49. 2:31AND SO THIS WOULD BE -12.
50. 2:34SO NOW LET'S GO BACK TO OUR IDENTITY,
51. 2:38AND WE SHOULD BE ABLE TO FIND ALL OF THESE VALUES NOW.
52. 2:41WE HAVE THE SINE OF ANGLE A - B = SINE A x COSINE B.
53. 2:51WELL, SINE A IS GIVEN AS 4/5,
54. 2:56COSINE B IS ALSO GIVEN AS - 5/13.
55. 3:00SINCE WE'RE USING A DIFFERENCE HERE
56. 3:01WE'LL HAVE A SUBTRACTION SIGN HERE.
57. 3:04NOW, THE COSINE OF ANGLE A
58. 3:05WE CAN USE THIS REFERENCE TRIANGLE HERE
59. 3:08ADJACENT OVER HYPOTENUSE, THAT'S -3/5.
60. 3:13AND FOR THE SINE OF ANGLE B
61. 3:14WE'LL USE THIS REFERENCE TRIANGLE OPPOSITE
62. 3:17OVER HYPOTENUSE OR -12/13.
63. 3:22NOW AGAIN, NOTICE WE HAVE A COMMON DENOMINATOR
64. 3:25OF 5 x 13 OR 65.
65. 3:29OUR FIRST NUMERATOR IS -20 - THIS WILL BE POSITIVE 36,
66. 3:38SO THE SINE OF A - B WILL EQUAL -56/65.
67. 3:45AND THAT'S WHAT WE WERE LOOKING FOR IN THIS PROBLEM.
68. 3:49LET'S TAKE A LOOK AT ANOTHER EXAMPLE NOW.
69. 3:51WE WANT TO DETERMINE THE EXACT VALUE OF SINE 150 DEGREES.
70. 3:56150 DEGREES IS NOT ONE OF THOSE NICE REFERENCE ANGLES,
71. 4:01BUT WE CAN USE A SUM OR DIFFERENCE OF REFERENCE ANGLES
72. 4:04TO OBTAIN 105 DEGREES.
73. 4:06105 DEGREES IS EQUAL TO 60 DEGREES + 45 DEGREES,
74. 4:14BOTH OF WHICH ARE REFERENCE ANGLES
75. 4:17THAT WE CAN USE IN THIS IDENTITY
76. 4:18TO DETERMINE THE EXACT VALUE OF SINE 105 DEGREES.
77. 4:22SO LET'S GO AHEAD AND SET IT UP.
78. 4:31SO SINCE THE SINE OF 105 DEGREES
79. 4:32EQUALS THE SINE OF 60 + 45 DEGREES,
80. 4:36OUR VALUE FOR A WILL BE 60 DEGREES
81. 4:38AND OUR VALUE FOR B WILL BE 45 DEGREES.
82. 4:43SO THIS WILL EQUAL THE SINE OF A x THE COSINE OF
83. 4:57+ COSINE OF A x THE SINE OF B.
84. 5:08AND WE'LL USE A UNIT CIRCLE TO FIND THESE VALUES
85. 5:11AND THEN FIND THEIR PRODUCTS AND THEN FIND THE SUM.
86. 5:14SO THE SINE OF 60 DEGREES,
87. 5:16REMEMBER SINE IS EQUAL TO Y ON THE UNIT CIRCLE.
88. 5:20SQUARE ROOT 3 OVER 2 TIMES THE COSINE OF 45 DEGREES
89. 5:26EQUAL TO THE X COORDINATE SQUARE ROOT 2 OVER 2
90. 5:30PLUS COSINE OF 60 WHICH IS EQUAL TO 1/2,
91. 5:37AND THE SINE OF 45 SQUARE ROOT 2 OVER 2.
92. 5:43SO WHAT WE HAVE FOUND IS THE SINE OF 105 DEGREES
93. 5:48IS EQUAL TO THE SUM OF THESE PRODUCTS.
94. 5:54NOTICE OUR DENOMINATOR IS 4 IN BOTH CASES.
95. 5:57HERE THE NUMERATOR IS THE SQUARE ROOT OF 6
96. 6:00+ THE SQUARE ROOT OF 2.
97. 6:06OKAY. I THINK I HAVE ONE MORE EXAMPLE.
98. 6:11NOW WE HAVE AN ANGLE THAT'S IN RADIANS,
99. 6:12SO IT MIGHT BE HELPFUL TO CONVERT THIS TO DEGREES
100. 6:15SO THAT WE CAN DETERMINE WHICH REFERENCE ANGLES TO USE.
101. 6:24THIS EQUALS -15 DEGREES.
102. 6:27SO WHAT WE CAN DO IS WE CAN USE 30 DEGREES - 45 DEGREES
103. 6:36AND THAT WILL GIVE US THE -15 DEGREES THAT WE NEED.
104. 6:39BUT LET'S BE CONSISTENT AND LET'S USE RADIANS.
105. 6:43SO 30 DEGREES IS THE SAME AS PI OVER 6 RADIANS,
106. 6:48AND 45 DEGREES IS THE SAME AS PI OVER 4 RADIANS.
107. 6:56SO WE'LL USE THIS FOR ANGLE A AND THIS FOR ANGLE B.
108. 6:59LET'S SET IT UP.
109. 7:03SO WE'LL HAVE THE SINE OF PI OVER 6 - PI OVER 4,
110. 7:07WHICH AGAIN WILL GIVE US THE -PI OVER 12.
111. 7:12SO WE'LL HAVE THE SINE OF A,
112. 7:13WHICH IS THE SINE OF PI OVER 6 x THE COSINE OF ANGLE B
113. 7:20WHICH IS PI OVER 4 - COSINE OF ANGLE A OR COSINE OF PI OVER 6
114. 7:30x THE SINE OF B WHICH IS PI OVER 4.
115. 7:39USING OUR UNIT CIRCLE,
116. 7:40THE SINE OF PI OVER 6 WOULD BE 1/2,
117. 7:46COSINE OF PI OVER 4 IS SQUARE ROOT 2 OVER 2,
118. 7:52COSINE OF PI OVER 6 WOULD BE SQUARE ROOT 3 OVER 2,
119. 7:57AND THE SINE OF PI OVER 4,
120. 7:59AGAIN, IS SQUARE ROOT 2 OVER 2.
121. 8:01OKAY. SO NOW WE HAVE THE VALUE OF SINE -PI OVER 12.
122. 8:06NOTICE WE HAVE A COMMON DENOMINATOR OF 4.
123. 8:10THIS NUMERATOR WOULD BE THE SQUARE ROOT OF 2 -
124. 8:14AND THIS IS THE SQUARE ROOT OF 6,
125. 8:16AND WE HAVE THE EXACT VALUE OF SINE -PI OVER 12.
126. 8:23OKAY. THAT'S PRETTY MUCH ALL WE HAVE TIME FOR.
127. 8:25I HOPE YOU FOUND THIS VIDEO HELPFUL.
128. 8:28THANK YOU FOR WATCHING.