{"message": {"transcript": [{"chunks": [{"end": 0.2, "start": 0.0, "text": "In"}, {"end": 0.56, "start": 0.2, "text": "this"}, {"end": 1.2, "start": 0.56, "text": "video,"}, {"end": 1.36, "start": 1.2, "text": "I"}, {"end": 1.36, "start": 1.36, "text": "want"}, {"end": 1.44, "start": 1.36, "text": "to"}, {"end": 1.64, "start": 1.44, "text": "give"}, {"end": 1.72, "start": 1.64, "text": "you"}, {"end": 1.84, "start": 1.72, "text": "the"}, {"end": 2.52, "start": 1.84, "text": "basics"}, {"end": 2.92, "start": 2.52, "text": "of"}, {"end": 4.36, "start": 2.92, "text": "trigonometry."}, {"end": 5.32, "start": 4.36, "text": "Trigonometry."}, {"end": 5.6, "start": 5.32, "text": "And"}, {"end": 5.88, "start": 5.6, "text": "it"}, {"end": 6.68, "start": 5.88, "text": "sounds"}, {"end": 7.12, "start": 6.68, "text": "like"}, {"end": 7.36, "start": 7.12, "text": "a"}, {"end": 7.44, "start": 7.36, "text": "very"}, {"end": 7.84, "start": 7.44, "text": "complicated"}, {"end": 7.84, "start": 7.84, "text": "topic,"}, {"end": 7.84, "start": 7.84, "text": "but"}, {"end": 7.88, "start": 7.84, "text": "you're"}, {"end": 8.2, "start": 7.88, "text": "going"}, {"end": 8.24, "start": 8.2, "text": "to"}, {"end": 8.36, "start": 8.24, "text": "see"}, {"end": 8.64, "start": 8.36, "text": "that"}, {"end": 8.84, "start": 8.64, "text": "it's"}, {"end": 9.56, "start": 8.84, "text": "really"}, {"end": 10.4, "start": 9.56, "text": "just"}, {"end": 10.8, "start": 10.4, "text": "the"}, {"end": 11.56, "start": 10.8, "text": "study"}, {"end": 11.6, "start": 11.56, "text": "of"}, {"end": 11.84, "start": 11.6, "text": "the"}, {"end": 12.68, "start": 11.84, "text": "ratios"}, {"end": 12.76, "start": 12.68, "text": "of"}, {"end": 13.48, "start": 12.76, "text": "sides"}, {"end": 13.64, "start": 13.48, "text": "of"}, {"end": 14.92, "start": 13.64, "text": "triangles."}, {"end": 15.08, "start": 14.92, "text": "The"}, {"end": 15.56, "start": 15.08, "text": "trig"}, {"end": 15.56, "start": 15.56, "text": "part"}, {"end": 15.6, "start": 15.56, "text": "of"}, {"end": 16.32, "start": 15.6, "text": "trigonometry"}, {"end": 16.64, "start": 16.32, "text": "literally"}, {"end": 17.4, "start": 16.64, "text": "means"}, {"end": 18.48, "start": 17.4, "text": "triangle,"}, {"end": 18.56, "start": 18.48, "text": "and"}, {"end": 18.8, "start": 18.56, "text": "the"}, {"end": 19.24, "start": 18.8, "text": "metri"}, {"end": 19.64, "start": 19.24, "text": "part"}, {"end": 20.08, "start": 19.64, "text": "literally"}, {"end": 20.72, "start": 20.08, "text": "means"}, {"end": 21.28, "start": 20.72, "text": "measure."}, {"end": 21.32, "start": 21.28, "text": "So"}, {"end": 21.4, "start": 21.32, "text": "let"}, {"end": 21.4, "start": 21.4, "text": "me"}, {"end": 21.4, "start": 21.4, "text": "just"}, {"end": 21.44, "start": 21.4, "text": "give"}, {"end": 21.76, "start": 21.44, "text": "you"}, {"end": 22.08, "start": 21.76, "text": "some"}, {"end": 22.64, "start": 22.08, "text": "examples"}, {"end": 23.08, "start": 22.64, "text": "here,"}, {"end": 23.52, "start": 23.08, "text": "and"}, {"end": 23.6, "start": 23.52, "text": "I"}, {"end": 24.04, "start": 23.6, "text": "think"}, {"end": 24.28, "start": 24.04, "text": "it'll"}, {"end": 24.52, "start": 24.28, "text": "make"}, {"end": 25.0, "start": 24.52, "text": "everything"}, {"end": 25.4, "start": 25.0, "text": "pretty"}, {"end": 25.76, "start": 25.4, "text": "clear."}, {"end": 25.76, "start": 25.76, "text": "So"}, {"end": 25.76, "start": 25.76, "text": "let"}, {"end": 26.2, "start": 25.76, "text": "me"}, {"end": 26.2, "start": 26.2, "text": "draw"}, {"end": 26.44, "start": 26.2, "text": "some"}, {"end": 26.6, "start": 26.44, "text": "right"}, {"end": 27.2, "start": 26.6, "text": "triangles."}, {"end": 27.44, "start": 27.2, "text": "Let"}, {"end": 27.44, "start": 27.44, "text": "me"}, {"end": 28.08, "start": 27.44, "text": "just"}, {"end": 28.44, "start": 28.08, "text": "draw"}, {"end": 28.68, "start": 28.44, "text": "one"}, {"end": 28.92, "start": 28.68, "text": "right"}, {"end": 29.96, "start": 28.92, "text": "triangle."}], "text": " In this video, I want to give you the basics of trigonometry. Trigonometry. And it sounds like a very complicated topic, but you're going to see that it's really just the study of the ratios of sides of triangles. The trig part of trigonometry literally means triangle, and the metri part literally means measure. So let me just give you some examples here, and I think it'll make everything pretty clear. So let me draw some right triangles. Let me just draw one right triangle."}, {"chunks": [{"end": 30.4, "start": 30.0, "text": "Is"}, {"end": 30.64, "start": 30.4, "text": "a"}, {"end": 30.92, "start": 30.64, "text": "right"}, {"end": 32.48, "start": 30.92, "text": "triangle."}, {"end": 32.76, "start": 32.48, "text": "When"}, {"end": 33.0, "start": 32.76, "text": "I"}, {"end": 33.48, "start": 33.0, "text": "say"}, {"end": 33.68, "start": 33.48, "text": "it's"}, {"end": 33.88, "start": 33.68, "text": "a"}, {"end": 34.28, "start": 33.88, "text": "right"}, {"end": 35.32, "start": 34.28, "text": "triangle,"}, {"end": 35.92, "start": 35.32, "text": "it's"}, {"end": 36.24, "start": 35.92, "text": "because"}, {"end": 36.24, "start": 36.24, "text": "one"}, {"end": 36.24, "start": 36.24, "text": "of"}, {"end": 36.24, "start": 36.24, "text": "the"}, {"end": 36.28, "start": 36.24, "text": "angles"}, {"end": 36.519999999999996, "start": 36.28, "text": "here"}, {"end": 36.88, "start": 36.519999999999996, "text": "is"}, {"end": 36.92, "start": 36.88, "text": "90"}, {"end": 38.04, "start": 36.92, "text": "degrees."}, {"end": 38.56, "start": 38.04, "text": "This"}, {"end": 39.0, "start": 38.56, "text": "right"}, {"end": 40.0, "start": 39.0, "text": "here"}, {"end": 40.12, "start": 40.0, "text": "is"}, {"end": 40.36, "start": 40.12, "text": "a"}, {"end": 40.64, "start": 40.36, "text": "right"}, {"end": 40.68, "start": 40.64, "text": "angle."}, {"end": 42.12, "start": 40.68, "text": "It"}, {"end": 42.84, "start": 42.12, "text": "is"}, {"end": 43.64, "start": 42.84, "text": "equal"}, {"end": 44.56, "start": 43.64, "text": "to"}, {"end": 45.480000000000004, "start": 44.56, "text": "90"}, {"end": 46.16, "start": 45.480000000000004, "text": "degrees."}, {"end": 46.16, "start": 46.16, "text": "And"}, {"end": 46.519999999999996, "start": 46.16, "text": "we'll"}, {"end": 47.0, "start": 46.519999999999996, "text": "talk"}, {"end": 47.28, "start": 47.0, "text": "about"}, {"end": 47.28, "start": 47.28, "text": "other"}, {"end": 47.68, "start": 47.28, "text": "ways"}, {"end": 47.96, "start": 47.68, "text": "to"}, {"end": 48.2, "start": 47.96, "text": "show"}, {"end": 48.32, "start": 48.2, "text": "the"}, {"end": 49.239999999999995, "start": 48.32, "text": "magnitude"}, {"end": 49.239999999999995, "start": 49.239999999999995, "text": "of"}, {"end": 49.239999999999995, "start": 49.239999999999995, "text": "angles"}, {"end": 49.239999999999995, "start": 49.239999999999995, "text": "in"}, {"end": 49.760000000000005, "start": 49.239999999999995, "text": "future"}, {"end": 49.760000000000005, "start": 49.760000000000005, "text": "videos."}, {"end": 49.760000000000005, "start": 49.760000000000005, "text": "So"}, {"end": 50.0, "start": 49.760000000000005, "text": "we"}, {"end": 50.56, "start": 50.0, "text": "have"}, {"end": 51.0, "start": 50.56, "text": "a"}, {"end": 51.56, "start": 51.0, "text": "90"}, {"end": 52.36, "start": 51.56, "text": "degree"}, {"end": 52.64, "start": 52.36, "text": "angle."}, {"end": 53.16, "start": 52.64, "text": "It's"}, {"end": 54.239999999999995, "start": 53.16, "text": "a"}, {"end": 54.56, "start": 54.239999999999995, "text": "right"}, {"end": 55.0, "start": 54.56, "text": "triangle."}, {"end": 55.32, "start": 55.0, "text": "Let"}, {"end": 55.44, "start": 55.32, "text": "me"}, {"end": 55.44, "start": 55.44, "text": "put"}, {"end": 55.879999999999995, "start": 55.44, "text": "some"}, {"end": 56.16, "start": 55.879999999999995, "text": "lengths"}, {"end": 56.16, "start": 56.16, "text": "to"}, {"end": 56.239999999999995, "start": 56.16, "text": "the"}, {"end": 56.68, "start": 56.239999999999995, "text": "sides"}, {"end": 56.879999999999995, "start": 56.68, "text": "here."}, {"end": 57.28, "start": 56.879999999999995, "text": "So"}, {"end": 57.879999999999995, "start": 57.28, "text": "this"}, {"end": 58.480000000000004, "start": 57.879999999999995, "text": "side"}, {"end": 59.0, "start": 58.480000000000004, "text": "over"}, {"end": 59.16, "start": 59.0, "text": "here"}, {"end": 59.480000000000004, "start": 59.16, "text": "is"}, {"end": 59.879999999999995, "start": 59.480000000000004, "text": "maybe"}, {"end": 59.96, "start": 59.879999999999995, "text": "3."}], "text": " Is a right triangle. When I say it's a right triangle, it's because one of the angles here is 90 degrees. This right here is a right angle. It is equal to 90 degrees. And we'll talk about other ways to show the magnitude of angles in future videos. So we have a 90 degree angle. It's a right triangle. Let me put some lengths to the sides here. So this side over here is maybe 3."}, {"chunks": [{"end": 60.36, "start": 60.0, "text": "This"}, {"end": 61.12, "start": 60.36, "text": "height"}, {"end": 61.16, "start": 61.12, "text": "right"}, {"end": 61.2, "start": 61.16, "text": "over"}, {"end": 61.64, "start": 61.2, "text": "there"}, {"end": 61.96, "start": 61.64, "text": "is"}, {"end": 62.96, "start": 61.96, "text": "3."}, {"end": 63.52, "start": 62.96, "text": "Maybe"}, {"end": 63.76, "start": 63.52, "text": "the"}, {"end": 64.24, "start": 63.76, "text": "base"}, {"end": 64.32, "start": 64.24, "text": "of"}, {"end": 64.32, "start": 64.32, "text": "the"}, {"end": 65.24, "start": 64.32, "text": "triangle"}, {"end": 66.32, "start": 65.24, "text": "right"}, {"end": 66.8, "start": 66.32, "text": "over"}, {"end": 67.12, "start": 66.8, "text": "here"}, {"end": 67.52, "start": 67.12, "text": "is"}, {"end": 67.92, "start": 67.52, "text": "4."}, {"end": 69.08, "start": 67.92, "text": "And"}, {"end": 69.8, "start": 69.08, "text": "then"}, {"end": 70.28, "start": 69.8, "text": "the"}, {"end": 70.96000000000001, "start": 70.28, "text": "hypotenuse"}, {"end": 71.16, "start": 70.96000000000001, "text": "of"}, {"end": 71.28, "start": 71.16, "text": "the"}, {"end": 71.88, "start": 71.28, "text": "triangle"}, {"end": 72.16, "start": 71.88, "text": "over"}, {"end": 72.6, "start": 72.16, "text": "here"}, {"end": 72.96000000000001, "start": 72.6, "text": "is"}, {"end": 73.64, "start": 72.96000000000001, "text": "5."}, {"end": 74.0, "start": 73.64, "text": "You"}, {"end": 74.64, "start": 74.0, "text": "only"}, {"end": 74.84, "start": 74.64, "text": "have"}, {"end": 74.84, "start": 74.84, "text": "a"}, {"end": 75.4, "start": 74.84, "text": "hypotenuse"}, {"end": 75.44, "start": 75.4, "text": "when"}, {"end": 75.64, "start": 75.44, "text": "you"}, {"end": 75.76, "start": 75.64, "text": "have"}, {"end": 75.8, "start": 75.76, "text": "a"}, {"end": 76.2, "start": 75.8, "text": "right"}, {"end": 76.76, "start": 76.2, "text": "triangle."}, {"end": 76.96000000000001, "start": 76.76, "text": "It"}, {"end": 77.24, "start": 76.96000000000001, "text": "is"}, {"end": 77.36, "start": 77.24, "text": "the"}, {"end": 77.92, "start": 77.36, "text": "side"}, {"end": 78.6, "start": 77.92, "text": "opposite"}, {"end": 78.6, "start": 78.6, "text": "the"}, {"end": 79.03999999999999, "start": 78.6, "text": "right"}, {"end": 79.28, "start": 79.03999999999999, "text": "angle,"}, {"end": 79.72, "start": 79.28, "text": "and"}, {"end": 80.03999999999999, "start": 79.72, "text": "it"}, {"end": 80.28, "start": 80.03999999999999, "text": "is"}, {"end": 80.52, "start": 80.28, "text": "the"}, {"end": 81.03999999999999, "start": 80.52, "text": "longest"}, {"end": 81.52, "start": 81.03999999999999, "text": "side"}, {"end": 81.84, "start": 81.52, "text": "of"}, {"end": 82.0, "start": 81.84, "text": "a"}, {"end": 82.0, "start": 82.0, "text": "right"}, {"end": 82.96000000000001, "start": 82.0, "text": "triangle."}, {"end": 83.12, "start": 82.96000000000001, "text": "So"}, {"end": 83.36, "start": 83.12, "text": "that"}, {"end": 83.56, "start": 83.36, "text": "right"}, {"end": 84.03999999999999, "start": 83.56, "text": "there"}, {"end": 84.56, "start": 84.03999999999999, "text": "is"}, {"end": 85.0, "start": 84.56, "text": "the"}, {"end": 86.36, "start": 85.0, "text": "hypotenuse."}, {"end": 86.68, "start": 86.36, "text": "You"}, {"end": 87.4, "start": 86.68, "text": "probably"}, {"end": 87.76, "start": 87.4, "text": "learned"}, {"end": 88.03999999999999, "start": 87.76, "text": "that"}, {"end": 88.44, "start": 88.03999999999999, "text": "already"}, {"end": 88.76, "start": 88.44, "text": "from"}, {"end": 89.96000000000001, "start": 88.76, "text": "geometry."}], "text": " This height right over there is 3. Maybe the base of the triangle right over here is 4. And then the hypotenuse of the triangle over here is 5. You only have a hypotenuse when you have a right triangle. It is the side opposite the right angle, and it is the longest side of a right triangle. So that right there is the hypotenuse. You probably learned that already from geometry."}, {"chunks": [{"end": 90.04, "start": 90.0, "text": "So"}, {"end": 90.04, "start": 90.04, "text": "you"}, {"end": 90.08, "start": 90.04, "text": "can"}, {"end": 90.68, "start": 90.08, "text": "verify"}, {"end": 90.72, "start": 90.68, "text": "that"}, {"end": 91.0, "start": 90.72, "text": "this"}, {"end": 91.24, "start": 91.0, "text": "right"}, {"end": 91.72, "start": 91.24, "text": "triangle,"}, {"end": 91.8, "start": 91.72, "text": "the"}, {"end": 92.24, "start": 91.8, "text": "sides"}, {"end": 92.36, "start": 92.24, "text": "work"}, {"end": 92.4, "start": 92.36, "text": "out."}, {"end": 92.4, "start": 92.4, "text": "We"}, {"end": 92.92, "start": 92.4, "text": "know"}, {"end": 93.08, "start": 92.92, "text": "from"}, {"end": 93.24, "start": 93.08, "text": "the"}, {"end": 94.0, "start": 93.24, "text": "Pythagorean"}, {"end": 94.8, "start": 94.0, "text": "theorem"}, {"end": 95.12, "start": 94.8, "text": "that"}, {"end": 95.4, "start": 95.12, "text": "3"}, {"end": 96.04, "start": 95.4, "text": "squared"}, {"end": 96.92, "start": 96.04, "text": "plus"}, {"end": 97.36, "start": 96.92, "text": "4"}, {"end": 98.68, "start": 97.36, "text": "squared"}, {"end": 99.24, "start": 98.68, "text": "has"}, {"end": 99.72, "start": 99.24, "text": "got"}, {"end": 100.0, "start": 99.72, "text": "to"}, {"end": 100.16, "start": 100.0, "text": "be"}, {"end": 100.2, "start": 100.16, "text": "equal"}, {"end": 100.48, "start": 100.2, "text": "to"}, {"end": 100.8, "start": 100.48, "text": "the"}, {"end": 101.12, "start": 100.8, "text": "length"}, {"end": 101.28, "start": 101.12, "text": "of"}, {"end": 101.84, "start": 101.28, "text": "the"}, {"end": 102.4, "start": 101.84, "text": "longest"}, {"end": 102.8, "start": 102.4, "text": "side."}, {"end": 102.84, "start": 102.8, "text": "The"}, {"end": 103.32, "start": 102.84, "text": "length"}, {"end": 103.32, "start": 103.32, "text": "of"}, {"end": 103.68, "start": 103.32, "text": "the"}, {"end": 104.2, "start": 103.68, "text": "hypotenuse"}, {"end": 104.64, "start": 104.2, "text": "squared"}, {"end": 104.8, "start": 104.64, "text": "is"}, {"end": 104.92, "start": 104.8, "text": "equal"}, {"end": 105.36, "start": 104.92, "text": "to"}, {"end": 105.76, "start": 105.36, "text": "5"}, {"end": 106.16, "start": 105.76, "text": "squared."}, {"end": 106.48, "start": 106.16, "text": "So"}, {"end": 106.72, "start": 106.48, "text": "you"}, {"end": 106.76, "start": 106.72, "text": "can"}, {"end": 107.44, "start": 106.76, "text": "verify"}, {"end": 107.44, "start": 107.44, "text": "that"}, {"end": 108.03999999999999, "start": 107.44, "text": "this"}, {"end": 108.36, "start": 108.03999999999999, "text": "works"}, {"end": 108.76, "start": 108.36, "text": "out."}, {"end": 108.84, "start": 108.76, "text": "This"}, {"end": 109.8, "start": 108.84, "text": "satisfies"}, {"end": 109.96000000000001, "start": 109.8, "text": "the"}, {"end": 110.96000000000001, "start": 109.96000000000001, "text": "Pythagorean"}, {"end": 111.2, "start": 110.96000000000001, "text": "theorem."}, {"end": 111.48, "start": 111.2, "text": "Now"}, {"end": 111.48, "start": 111.48, "text": "with"}, {"end": 112.0, "start": 111.48, "text": "that"}, {"end": 112.4, "start": 112.0, "text": "out"}, {"end": 112.6, "start": 112.4, "text": "of"}, {"end": 112.6, "start": 112.6, "text": "the"}, {"end": 112.8, "start": 112.6, "text": "way,"}, {"end": 113.12, "start": 112.8, "text": "let's"}, {"end": 113.64, "start": 113.12, "text": "learn"}, {"end": 113.96000000000001, "start": 113.64, "text": "a"}, {"end": 114.64, "start": 113.96000000000001, "text": "little"}, {"end": 114.88, "start": 114.64, "text": "bit"}, {"end": 115.12, "start": 114.88, "text": "of"}, {"end": 115.52, "start": 115.12, "text": "trigonometry."}, {"end": 116.32, "start": 115.52, "text": "So"}, {"end": 116.36, "start": 116.32, "text": "the"}, {"end": 116.96000000000001, "start": 116.36, "text": "core"}, {"end": 117.88, "start": 116.96000000000001, "text": "functions"}, {"end": 118.12, "start": 117.88, "text": "of"}, {"end": 118.68, "start": 118.12, "text": "trigonometry,"}, {"end": 118.84, "start": 118.68, "text": "we're"}, {"end": 118.84, "start": 118.84, "text": "going"}, {"end": 118.88, "start": 118.84, "text": "to"}, {"end": 118.92, "start": 118.88, "text": "learn"}, {"end": 119.03999999999999, "start": 118.92, "text": "a"}, {"end": 119.16, "start": 119.03999999999999, "text": "little"}, {"end": 119.16, "start": 119.16, "text": "bit"}, {"end": 119.32, "start": 119.16, "text": "more"}, {"end": 119.52, "start": 119.32, "text": "about"}, {"end": 119.96000000000001, "start": 119.52, "text": "what"}], "text": " So you can verify that this right triangle, the sides work out. We know from the Pythagorean theorem that 3 squared plus 4 squared has got to be equal to the length of the longest side. The length of the hypotenuse squared is equal to 5 squared. So you can verify that this works out. This satisfies the Pythagorean theorem. Now with that out of the way, let's learn a little bit of trigonometry. So the core functions of trigonometry, we're going to learn a little bit more about what"}, {"chunks": [{"end": 120.52, "start": 120.0, "text": "There"}, {"end": 121.28, "start": 120.52, "text": "is"}, {"end": 121.92, "start": 121.28, "text": "the"}, {"end": 123.12, "start": 121.92, "text": "sine"}, {"end": 124.08, "start": 123.12, "text": "function."}, {"end": 124.72, "start": 124.08, "text": "There"}, {"end": 125.28, "start": 124.72, "text": "is"}, {"end": 125.48, "start": 125.28, "text": "the"}, {"end": 126.24, "start": 125.48, "text": "cosine"}, {"end": 126.92, "start": 126.24, "text": "function."}, {"end": 127.56, "start": 126.92, "text": "And"}, {"end": 128.16, "start": 127.56, "text": "there"}, {"end": 128.92, "start": 128.16, "text": "is"}, {"end": 129.16, "start": 128.92, "text": "the"}, {"end": 129.64, "start": 129.16, "text": "tangent"}, {"end": 130.16, "start": 129.64, "text": "function."}, {"end": 130.4, "start": 130.16, "text": "And"}, {"end": 131.0, "start": 130.4, "text": "you"}, {"end": 131.28, "start": 131.0, "text": "write"}, {"end": 131.84, "start": 131.28, "text": "sine,"}, {"end": 132.44, "start": 131.84, "text": "cos,"}, {"end": 132.52, "start": 132.44, "text": "or"}, {"end": 133.16, "start": 132.52, "text": "S-I-N,"}, {"end": 133.88, "start": 133.16, "text": "C-O-S,"}, {"end": 133.92, "start": 133.88, "text": "and"}, {"end": 134.44, "start": 133.92, "text": "tan"}, {"end": 134.88, "start": 134.44, "text": "for"}, {"end": 135.2, "start": 134.88, "text": "short."}, {"end": 135.2, "start": 135.2, "text": "And"}, {"end": 136.04, "start": 135.2, "text": "these"}, {"end": 136.32, "start": 136.04, "text": "really"}, {"end": 136.96, "start": 136.32, "text": "just"}, {"end": 137.48, "start": 136.96, "text": "specify,"}, {"end": 138.52, "start": 137.48, "text": "for"}, {"end": 138.68, "start": 138.52, "text": "any"}, {"end": 138.68, "start": 138.68, "text": "angle"}, {"end": 138.68, "start": 138.68, "text": "in"}, {"end": 138.96, "start": 138.68, "text": "this"}, {"end": 139.68, "start": 138.96, "text": "triangle,"}, {"end": 140.07999999999998, "start": 139.68, "text": "it'll"}, {"end": 140.8, "start": 140.07999999999998, "text": "specify"}, {"end": 140.96, "start": 140.8, "text": "the"}, {"end": 141.68, "start": 140.96, "text": "ratios"}, {"end": 141.8, "start": 141.68, "text": "of"}, {"end": 142.2, "start": 141.8, "text": "certain"}, {"end": 142.64, "start": 142.2, "text": "sides."}, {"end": 143.0, "start": 142.64, "text": "So"}, {"end": 143.04, "start": 143.0, "text": "let"}, {"end": 143.04, "start": 143.04, "text": "me"}, {"end": 143.32, "start": 143.04, "text": "just"}, {"end": 143.72, "start": 143.32, "text": "write"}, {"end": 143.72, "start": 143.72, "text": "something"}, {"end": 143.72, "start": 143.72, "text": "out."}, {"end": 143.72, "start": 143.72, "text": "And"}, {"end": 143.72, "start": 143.72, "text": "this"}, {"end": 143.8, "start": 143.72, "text": "is"}, {"end": 143.88, "start": 143.8, "text": "a"}, {"end": 144.16, "start": 143.88, "text": "little"}, {"end": 144.44, "start": 144.16, "text": "bit"}, {"end": 144.6, "start": 144.44, "text": "of"}, {"end": 145.07999999999998, "start": 144.6, "text": "a"}, {"end": 145.4, "start": 145.07999999999998, "text": "mnemonic"}, {"end": 145.64, "start": 145.4, "text": "here,"}, {"end": 145.8, "start": 145.64, "text": "so"}, {"end": 146.04, "start": 145.8, "text": "something"}, {"end": 146.48, "start": 146.04, "text": "just"}, {"end": 146.68, "start": 146.48, "text": "to"}, {"end": 146.92000000000002, "start": 146.68, "text": "help"}, {"end": 146.96, "start": 146.92000000000002, "text": "you"}, {"end": 147.6, "start": 146.96, "text": "remember"}, {"end": 147.76, "start": 147.6, "text": "the"}, {"end": 148.56, "start": 147.76, "text": "definitions"}, {"end": 149.07999999999998, "start": 148.56, "text": "of"}, {"end": 149.48, "start": 149.07999999999998, "text": "these"}, {"end": 149.96, "start": 149.48, "text": "functions."}], "text": " There is the sine function. There is the cosine function. And there is the tangent function. And you write sine, cos, or S-I-N, C-O-S, and tan for short. And these really just specify, for any angle in this triangle, it'll specify the ratios of certain sides. So let me just write something out. And this is a little bit of a mnemonic here, so something just to help you remember the definitions of these functions."}, {"chunks": [{"end": 150.44, "start": 150.0, "text": "But"}, {"end": 150.76, "start": 150.44, "text": "I'm"}, {"end": 151.2, "start": 150.76, "text": "going"}, {"end": 151.68, "start": 151.2, "text": "to"}, {"end": 152.16, "start": 151.68, "text": "write"}, {"end": 153.32, "start": 152.16, "text": "down"}, {"end": 154.08, "start": 153.32, "text": "something."}, {"end": 154.08, "start": 154.08, "text": "It's"}, {"end": 155.6, "start": 154.08, "text": "called"}, {"end": 156.48, "start": 155.6, "text": "so-ca-toa."}, {"end": 157.12, "start": 156.48, "text": "And"}, {"end": 158.48, "start": 157.12, "text": "you'll"}, {"end": 158.84, "start": 158.48, "text": "be"}, {"end": 159.16, "start": 158.84, "text": "amazed"}, {"end": 159.92, "start": 159.16, "text": "how"}, {"end": 160.64, "start": 159.92, "text": "far"}, {"end": 160.96, "start": 160.64, "text": "this"}, {"end": 161.24, "start": 160.96, "text": "mnemonic"}, {"end": 161.36, "start": 161.24, "text": "will"}, {"end": 161.68, "start": 161.36, "text": "take"}, {"end": 161.8, "start": 161.68, "text": "you"}, {"end": 161.8, "start": 161.8, "text": "in"}, {"end": 162.52, "start": 161.8, "text": "trigonometry."}, {"end": 162.52, "start": 162.52, "text": "So"}, {"end": 162.96, "start": 162.52, "text": "we"}, {"end": 163.0, "start": 162.96, "text": "have"}, {"end": 163.88, "start": 163.0, "text": "so-ca-toa."}, {"end": 164.2, "start": 163.88, "text": "And"}, {"end": 164.76, "start": 164.2, "text": "what"}, {"end": 165.36, "start": 164.76, "text": "this"}, {"end": 166.07999999999998, "start": 165.36, "text": "tells"}, {"end": 166.36, "start": 166.07999999999998, "text": "us"}, {"end": 166.52, "start": 166.36, "text": "is,"}, {"end": 166.68, "start": 166.52, "text": "so"}, {"end": 167.24, "start": 166.68, "text": "so"}, {"end": 168.12, "start": 167.24, "text": "tells"}, {"end": 168.88, "start": 168.12, "text": "us"}, {"end": 169.4, "start": 168.88, "text": "that"}, {"end": 170.04, "start": 169.4, "text": "sine"}, {"end": 170.36, "start": 170.04, "text": "is"}, {"end": 170.8, "start": 170.36, "text": "equal"}, {"end": 171.36, "start": 170.8, "text": "to"}, {"end": 172.0, "start": 171.36, "text": "opposite"}, {"end": 172.44, "start": 172.0, "text": "over"}, {"end": 173.24, "start": 172.44, "text": "hypotenuse."}, {"end": 173.64, "start": 173.24, "text": "It's"}, {"end": 174.48, "start": 173.64, "text": "telling"}, {"end": 175.36, "start": 174.48, "text": "us,"}, {"end": 175.6, "start": 175.36, "text": "and"}, {"end": 176.12, "start": 175.6, "text": "this"}, {"end": 176.96, "start": 176.12, "text": "won't"}, {"end": 178.52, "start": 176.96, "text": "make"}, {"end": 179.28, "start": 178.52, "text": "a"}, {"end": 179.96, "start": 179.28, "text": "lot"}], "text": " But I'm going to write down something. It's called so-ca-toa. And you'll be amazed how far this mnemonic will take you in trigonometry. So we have so-ca-toa. And what this tells us is, so so tells us that sine is equal to opposite over hypotenuse. It's telling us, and this won't make a lot"}, {"chunks": [{"end": 180.4, "start": 180.0, "text": "Let's"}, {"end": 180.84, "start": 180.4, "text": "sense"}, {"end": 181.2, "start": 180.84, "text": "just"}, {"end": 181.4, "start": 181.2, "text": "yet."}, {"end": 181.68, "start": 181.4, "text": "I'll"}, {"end": 181.68, "start": 181.68, "text": "do"}, {"end": 181.68, "start": 181.68, "text": "it"}, {"end": 181.68, "start": 181.68, "text": "in"}, {"end": 181.68, "start": 181.68, "text": "a"}, {"end": 181.72, "start": 181.68, "text": "little"}, {"end": 181.8, "start": 181.72, "text": "bit"}, {"end": 182.04, "start": 181.8, "text": "more"}, {"end": 182.4, "start": 182.04, "text": "detail"}, {"end": 182.4, "start": 182.4, "text": "in"}, {"end": 182.56, "start": 182.4, "text": "a"}, {"end": 183.04, "start": 182.56, "text": "second."}, {"end": 183.44, "start": 183.04, "text": "And"}, {"end": 183.76, "start": 183.44, "text": "then"}, {"end": 184.64, "start": 183.76, "text": "cosine"}, {"end": 184.84, "start": 184.64, "text": "is"}, {"end": 185.04, "start": 184.84, "text": "equal"}, {"end": 185.48, "start": 185.04, "text": "to"}, {"end": 186.96, "start": 185.48, "text": "adjacent"}, {"end": 187.56, "start": 186.96, "text": "over"}, {"end": 189.2, "start": 187.56, "text": "hypotenuse."}, {"end": 189.8, "start": 189.2, "text": "And"}, {"end": 190.08, "start": 189.8, "text": "then"}, {"end": 190.36, "start": 190.08, "text": "you"}, {"end": 191.08, "start": 190.36, "text": "finally"}, {"end": 191.36, "start": 191.08, "text": "have"}, {"end": 191.68, "start": 191.36, "text": "tangent"}, {"end": 192.6, "start": 191.68, "text": "is"}, {"end": 192.76, "start": 192.6, "text": "equal"}, {"end": 193.36, "start": 192.76, "text": "to"}, {"end": 195.44, "start": 193.36, "text": "opposite"}, {"end": 195.68, "start": 195.44, "text": "over"}, {"end": 196.6, "start": 195.68, "text": "adjacent."}, {"end": 197.4, "start": 196.6, "text": "So"}, {"end": 198.04, "start": 197.4, "text": "you're"}, {"end": 198.88, "start": 198.04, "text": "probably"}, {"end": 199.24, "start": 198.88, "text": "saying,"}, {"end": 199.48, "start": 199.24, "text": "hey"}, {"end": 199.88, "start": 199.48, "text": "Sal,"}, {"end": 200.12, "start": 199.88, "text": "what"}, {"end": 200.48, "start": 200.12, "text": "is"}, {"end": 200.48, "start": 200.48, "text": "all"}, {"end": 201.2, "start": 200.48, "text": "this"}, {"end": 201.76, "start": 201.2, "text": "opposite"}, {"end": 202.68, "start": 201.76, "text": "hypotenuse"}, {"end": 203.68, "start": 202.68, "text": "adjacent?"}, {"end": 204.04, "start": 203.68, "text": "What"}, {"end": 204.36, "start": 204.04, "text": "are"}, {"end": 204.84, "start": 204.36, "text": "we"}, {"end": 205.88, "start": 204.84, "text": "talking"}, {"end": 206.4, "start": 205.88, "text": "about?"}, {"end": 206.96, "start": 206.4, "text": "Well,"}, {"end": 207.48, "start": 206.96, "text": "let's"}, {"end": 208.36, "start": 207.48, "text": "take"}, {"end": 208.52, "start": 208.36, "text": "an"}, {"end": 209.07999999999998, "start": 208.52, "text": "angle"}, {"end": 209.96, "start": 209.07999999999998, "text": "here."}], "text": " Let's sense just yet. I'll do it in a little bit more detail in a second. And then cosine is equal to adjacent over hypotenuse. And then you finally have tangent is equal to opposite over adjacent. So you're probably saying, hey Sal, what is all this opposite hypotenuse adjacent? What are we talking about? Well, let's take an angle here."}, {"chunks": [{"end": 210.32, "start": 210.0, "text": "Let's"}, {"end": 210.84, "start": 210.32, "text": "say"}, {"end": 210.88, "start": 210.84, "text": "that"}, {"end": 211.32, "start": 210.88, "text": "this"}, {"end": 211.48, "start": 211.32, "text": "angle"}, {"end": 211.88, "start": 211.48, "text": "right"}, {"end": 212.84, "start": 211.88, "text": "over"}, {"end": 213.76, "start": 212.84, "text": "here"}, {"end": 214.2, "start": 213.76, "text": "is"}, {"end": 214.6, "start": 214.2, "text": "theta"}, {"end": 214.64, "start": 214.6, "text": "between"}, {"end": 215.0, "start": 214.64, "text": "the"}, {"end": 215.44, "start": 215.0, "text": "side"}, {"end": 215.76, "start": 215.44, "text": "of"}, {"end": 216.72, "start": 215.76, "text": "length"}, {"end": 216.92, "start": 216.72, "text": "4"}, {"end": 217.08, "start": 216.92, "text": "and"}, {"end": 217.08, "start": 217.08, "text": "the"}, {"end": 217.08, "start": 217.08, "text": "side"}, {"end": 217.08, "start": 217.08, "text": "of"}, {"end": 217.12, "start": 217.08, "text": "length"}, {"end": 217.56, "start": 217.12, "text": "5."}, {"end": 217.96, "start": 217.56, "text": "This"}, {"end": 218.08, "start": 217.96, "text": "angle"}, {"end": 218.56, "start": 218.08, "text": "right"}, {"end": 218.88, "start": 218.56, "text": "here"}, {"end": 218.96, "start": 218.88, "text": "is"}, {"end": 219.28, "start": 218.96, "text": "theta."}, {"end": 219.36, "start": 219.28, "text": "So"}, {"end": 219.72, "start": 219.36, "text": "let's"}, {"end": 220.4, "start": 219.72, "text": "figure"}, {"end": 220.52, "start": 220.4, "text": "out"}, {"end": 220.64, "start": 220.52, "text": "what"}, {"end": 220.76, "start": 220.64, "text": "the"}, {"end": 221.36, "start": 220.76, "text": "sine"}, {"end": 221.36, "start": 221.36, "text": "of"}, {"end": 221.88, "start": 221.36, "text": "theta,"}, {"end": 222.0, "start": 221.88, "text": "the"}, {"end": 222.6, "start": 222.0, "text": "cosine"}, {"end": 222.64, "start": 222.6, "text": "of"}, {"end": 223.12, "start": 222.64, "text": "theta,"}, {"end": 223.24, "start": 223.12, "text": "and"}, {"end": 223.28, "start": 223.24, "text": "what"}, {"end": 223.4, "start": 223.28, "text": "the"}, {"end": 224.12, "start": 223.4, "text": "tangent"}, {"end": 224.36, "start": 224.12, "text": "of"}, {"end": 225.0, "start": 224.36, "text": "theta"}, {"end": 225.4, "start": 225.0, "text": "are."}, {"end": 225.6, "start": 225.4, "text": "So"}, {"end": 225.84, "start": 225.6, "text": "if"}, {"end": 225.84, "start": 225.84, "text": "we"}, {"end": 225.84, "start": 225.84, "text": "want"}, {"end": 226.8, "start": 225.84, "text": "to"}, {"end": 227.56, "start": 226.8, "text": "first"}, {"end": 228.36, "start": 227.56, "text": "focus"}, {"end": 228.6, "start": 228.36, "text": "on"}, {"end": 229.4, "start": 228.6, "text": "the"}, {"end": 230.12, "start": 229.4, "text": "sine"}, {"end": 230.44, "start": 230.12, "text": "of"}, {"end": 230.84, "start": 230.44, "text": "theta,"}, {"end": 231.07999999999998, "start": 230.84, "text": "we"}, {"end": 231.68, "start": 231.07999999999998, "text": "just"}, {"end": 231.84, "start": 231.68, "text": "have"}, {"end": 231.84, "start": 231.84, "text": "to"}, {"end": 232.07999999999998, "start": 231.84, "text": "remember,"}, {"end": 232.72, "start": 232.07999999999998, "text": "soh"}, {"end": 233.32, "start": 232.72, "text": "ka"}, {"end": 233.64, "start": 233.32, "text": "toa."}, {"end": 234.12, "start": 233.64, "text": "Sine"}, {"end": 234.4, "start": 234.12, "text": "is"}, {"end": 235.12, "start": 234.4, "text": "opposite"}, {"end": 235.56, "start": 235.12, "text": "over"}, {"end": 235.84, "start": 235.56, "text": "hypotenuse."}, {"end": 236.2, "start": 235.84, "text": "So"}, {"end": 236.56, "start": 236.2, "text": "sine"}, {"end": 236.92000000000002, "start": 236.56, "text": "of"}, {"end": 237.6, "start": 236.92000000000002, "text": "theta"}, {"end": 238.07999999999998, "start": 237.6, "text": "is"}, {"end": 238.4, "start": 238.07999999999998, "text": "equal"}, {"end": 238.72, "start": 238.4, "text": "to"}, {"end": 239.4, "start": 238.72, "text": "the"}, {"end": 239.96, "start": 239.4, "text": "opposite."}], "text": " Let's say that this angle right over here is theta between the side of length 4 and the side of length 5. This angle right here is theta. So let's figure out what the sine of theta, the cosine of theta, and what the tangent of theta are. So if we want to first focus on the sine of theta, we just have to remember, soh ka toa. Sine is opposite over hypotenuse. So sine of theta is equal to the opposite."}, {"chunks": [{"end": 240.12, "start": 240.0, "text": "So"}, {"end": 240.52, "start": 240.12, "text": "what's"}, {"end": 240.6, "start": 240.52, "text": "the"}, {"end": 241.0, "start": 240.6, "text": "opposite"}, {"end": 241.6, "start": 241.0, "text": "side"}, {"end": 241.84, "start": 241.6, "text": "to"}, {"end": 242.24, "start": 241.84, "text": "the"}, {"end": 242.52, "start": 242.24, "text": "angle?"}, {"end": 242.72, "start": 242.52, "text": "So"}, {"end": 242.92, "start": 242.72, "text": "this"}, {"end": 243.24, "start": 242.92, "text": "is"}, {"end": 243.68, "start": 243.24, "text": "our"}, {"end": 243.84, "start": 243.68, "text": "angle"}, {"end": 244.2, "start": 243.84, "text": "right"}, {"end": 244.52, "start": 244.2, "text": "here."}, {"end": 245.12, "start": 244.52, "text": "The"}, {"end": 245.52, "start": 245.12, "text": "opposite"}, {"end": 245.96, "start": 245.52, "text": "side,"}, {"end": 246.28, "start": 245.96, "text": "if"}, {"end": 246.28, "start": 246.28, "text": "you"}, {"end": 246.28, "start": 246.28, "text": "just"}, {"end": 246.28, "start": 246.28, "text": "go"}, {"end": 246.56, "start": 246.28, "text": "the"}, {"end": 246.96, "start": 246.56, "text": "opposite"}, {"end": 247.44, "start": 246.96, "text": "side,"}, {"end": 247.84, "start": 247.44, "text": "so"}, {"end": 248.72, "start": 247.84, "text": "not"}, {"end": 248.88, "start": 248.72, "text": "one"}, {"end": 248.96, "start": 248.88, "text": "of"}, {"end": 249.6, "start": 248.96, "text": "the"}, {"end": 250.08, "start": 249.6, "text": "sides"}, {"end": 250.6, "start": 250.08, "text": "that"}, {"end": 250.84, "start": 250.6, "text": "are"}, {"end": 251.24, "start": 250.84, "text": "kind"}, {"end": 251.32, "start": 251.24, "text": "of"}, {"end": 251.84, "start": 251.32, "text": "adjacent"}, {"end": 252.2, "start": 251.84, "text": "to"}, {"end": 252.24, "start": 252.2, "text": "the"}, {"end": 252.4, "start": 252.24, "text": "angle,"}, {"end": 252.8, "start": 252.4, "text": "the"}, {"end": 253.12, "start": 252.8, "text": "opposite"}, {"end": 253.8, "start": 253.12, "text": "side"}, {"end": 254.04, "start": 253.8, "text": "is"}, {"end": 254.2, "start": 254.04, "text": "the"}, {"end": 255.0, "start": 254.2, "text": "3."}, {"end": 255.32, "start": 255.0, "text": "It's"}, {"end": 255.52, "start": 255.32, "text": "opening"}, {"end": 256.0, "start": 255.52, "text": "onto"}, {"end": 256.64, "start": 256.0, "text": "that"}, {"end": 256.92, "start": 256.64, "text": "3."}, {"end": 256.92, "start": 256.92, "text": "So"}, {"end": 257.0, "start": 256.92, "text": "the"}, {"end": 257.4, "start": 257.0, "text": "opposite"}, {"end": 257.84, "start": 257.4, "text": "side"}, {"end": 258.2, "start": 257.84, "text": "is"}, {"end": 258.8, "start": 258.2, "text": "3."}, {"end": 259.08, "start": 258.8, "text": "And"}, {"end": 259.36, "start": 259.08, "text": "then"}, {"end": 259.84, "start": 259.36, "text": "what's"}, {"end": 260.24, "start": 259.84, "text": "the"}, {"end": 261.08, "start": 260.24, "text": "hypotenuse?"}, {"end": 261.32, "start": 261.08, "text": "The"}, {"end": 262.0, "start": 261.32, "text": "hypotenuse,"}, {"end": 262.0, "start": 262.0, "text": "well,"}, {"end": 262.0, "start": 262.0, "text": "we"}, {"end": 262.28, "start": 262.0, "text": "already"}, {"end": 262.44, "start": 262.28, "text": "know."}, {"end": 262.52, "start": 262.44, "text": "The"}, {"end": 263.12, "start": 262.52, "text": "hypotenuse"}, {"end": 263.32, "start": 263.12, "text": "here"}, {"end": 263.6, "start": 263.32, "text": "is"}, {"end": 264.28, "start": 263.6, "text": "5."}, {"end": 264.68, "start": 264.28, "text": "So"}, {"end": 265.0, "start": 264.68, "text": "it's"}, {"end": 265.4, "start": 265.0, "text": "3"}, {"end": 265.48, "start": 265.4, "text": "over"}, {"end": 266.28, "start": 265.48, "text": "5."}, {"end": 266.8, "start": 266.28, "text": "The"}, {"end": 267.32, "start": 266.8, "text": "sine"}, {"end": 267.32, "start": 267.32, "text": "of"}, {"end": 267.32, "start": 267.32, "text": "theta"}, {"end": 267.32, "start": 267.32, "text": "is"}, {"end": 267.36, "start": 267.32, "text": "3"}, {"end": 267.84, "start": 267.36, "text": "fifths."}, {"end": 267.84, "start": 267.84, "text": "So"}, {"end": 267.84, "start": 267.84, "text": "someone"}, {"end": 268.56, "start": 267.84, "text": "says,"}, {"end": 269.0, "start": 268.56, "text": "hey,"}, {"end": 269.2, "start": 269.0, "text": "what's"}, {"end": 269.6, "start": 269.2, "text": "the"}, {"end": 269.96, "start": 269.6, "text": "sine"}], "text": " So what's the opposite side to the angle? So this is our angle right here. The opposite side, if you just go the opposite side, so not one of the sides that are kind of adjacent to the angle, the opposite side is the 3. It's opening onto that 3. So the opposite side is 3. And then what's the hypotenuse? The hypotenuse, well, we already know. The hypotenuse here is 5. So it's 3 over 5. The sine of theta is 3 fifths. So someone says, hey, what's the sine"}, {"chunks": [{"end": 270.56, "start": 270.0, "text": "So"}, {"end": 270.56, "start": 270.56, "text": "what"}, {"end": 270.56, "start": 270.56, "text": "do"}, {"end": 270.56, "start": 270.56, "text": "you"}, {"end": 270.56, "start": 270.56, "text": "think"}, {"end": 270.56, "start": 270.56, "text": "about"}, {"end": 270.64, "start": 270.56, "text": "that?"}, {"end": 270.88, "start": 270.64, "text": "It's"}, {"end": 271.48, "start": 270.88, "text": "3"}, {"end": 272.24, "start": 271.48, "text": "fifths."}, {"end": 272.24, "start": 272.24, "text": "And"}, {"end": 272.24, "start": 272.24, "text": "I'm"}, {"end": 272.24, "start": 272.24, "text": "going"}, {"end": 272.24, "start": 272.24, "text": "to"}, {"end": 272.24, "start": 272.24, "text": "show"}, {"end": 272.24, "start": 272.24, "text": "you"}, {"end": 272.4, "start": 272.24, "text": "in"}, {"end": 272.88, "start": 272.4, "text": "a"}, {"end": 273.64, "start": 272.88, "text": "second"}, {"end": 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"start": 283.8, "text": "were"}, {"end": 284.08, "start": 283.96, "text": "a"}, {"end": 284.52, "start": 284.08, "text": "larger"}, {"end": 285.0, "start": 284.52, "text": "triangle"}, {"end": 285.12, "start": 285.0, "text": "or"}, {"end": 285.32, "start": 285.12, "text": "a"}, {"end": 285.72, "start": 285.32, "text": "smaller"}, {"end": 285.88, "start": 285.72, "text": "one."}, {"end": 286.28, "start": 285.88, "text": "So"}, {"end": 286.64, "start": 286.28, "text": "I'll"}, {"end": 286.84, "start": 286.64, "text": "show"}, {"end": 286.84, "start": 286.84, "text": "you"}, {"end": 286.84, "start": 286.84, "text": "that"}, {"end": 286.88, "start": 286.84, "text": "in"}, {"end": 286.96, "start": 286.88, "text": "a"}, {"end": 287.4, "start": 286.96, "text": "second."}, {"end": 287.76, "start": 287.4, "text": "But"}, {"end": 288.2, "start": 287.76, "text": "let's"}, {"end": 288.36, "start": 288.2, "text": "go"}, {"end": 288.92, "start": 288.36, "text": "through"}, {"end": 289.32, "start": 288.92, "text": "all"}, {"end": 289.6, "start": 289.32, "text": "of"}, {"end": 289.76, "start": 289.6, "text": "the"}, {"end": 289.76, "start": 289.76, "text": "trig"}, {"end": 290.44, "start": 289.76, "text": "functions."}, {"end": 291.32, "start": 290.44, "text": "Let's"}, {"end": 291.64, "start": 291.32, "text": "think"}, {"end": 291.88, "start": 291.64, "text": "about"}, {"end": 291.88, "start": 291.88, "text": "what"}, {"end": 292.0, "start": 291.88, "text": "the"}, {"end": 292.6, "start": 292.0, "text": "cosine"}, {"end": 292.72, "start": 292.6, "text": "of"}, {"end": 292.76, "start": 292.72, "text": "theta"}, {"end": 293.6, "start": 292.76, "text": "is."}, {"end": 294.8, "start": 293.6, "text": "Cosine"}, {"end": 295.04, "start": 294.8, "text": "is"}, {"end": 296.12, "start": 295.04, "text": "adjacent"}, {"end": 296.64, "start": 296.12, "text": "over"}, {"end": 297.0, "start": 296.64, "text": "hypotenuse."}, {"end": 297.44, "start": 297.0, "text": "So"}, {"end": 297.6, "start": 297.44, "text": "remember,"}, {"end": 297.96, "start": 297.6, "text": "let"}, {"end": 298.64, "start": 297.96, "text": "me"}, {"end": 299.6, "start": 298.64, "text": "label"}, {"end": 299.96, "start": 299.6, "text": "them."}], "text": " So what do you think about that? It's 3 fifths. And I'm going to show you in a second that the sine of theta of, if this angle is a certain angle, it's always going to be 3 fifths. The ratio of the opposite to the hypotenuse is always going to be the same, even if the actual triangle were a larger triangle or a smaller one. So I'll show you that in a second. But let's go through all of the trig functions. Let's think about what the cosine of theta is. Cosine is adjacent over hypotenuse. So remember, let me label them."}, {"chunks": [{"end": 300.6, "start": 300.0, "text": "We"}, {"end": 300.8, "start": 300.6, "text": "already"}, {"end": 300.8, "start": 300.8, "text": "figured"}, {"end": 300.8, "start": 300.8, "text": "out"}, {"end": 300.84, "start": 300.8, "text": "that"}, {"end": 301.0, "start": 300.84, "text": "the"}, {"end": 301.36, "start": 301.0, "text": "3"}, {"end": 301.8, "start": 301.36, "text": "was"}, {"end": 301.88, "start": 301.8, "text": "the"}, {"end": 302.28, "start": 301.88, "text": "opposite"}, {"end": 303.2, "start": 302.28, "text": "side."}, {"end": 303.64, "start": 303.2, "text": "This"}, {"end": 303.72, "start": 303.64, "text": "is"}, {"end": 303.8, "start": 303.72, "text": "the"}, {"end": 304.2, "start": 303.8, "text": "opposite"}, {"end": 304.76, "start": 304.2, "text": "side."}, {"end": 305.08, "start": 304.76, "text": "And"}, {"end": 305.08, "start": 305.08, "text": "only"}, {"end": 305.08, "start": 305.08, "text": "when"}, {"end": 305.32, 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"very"}, {"end": 324.36, "start": 324.08, "text": "clear."}, {"end": 324.64, "start": 324.36, "text": "This"}, {"end": 325.04, "start": 324.64, "text": "only"}, {"end": 325.52, "start": 325.04, "text": "applies"}, {"end": 325.68, "start": 325.52, "text": "to"}, {"end": 326.2, "start": 325.68, "text": "this"}, {"end": 326.44, "start": 326.2, "text": "angle."}, {"end": 326.44, "start": 326.44, "text": "If"}, {"end": 326.44, "start": 326.44, "text": "we're"}, {"end": 326.76, "start": 326.44, "text": "talking"}, {"end": 326.88, "start": 326.76, "text": "about"}, {"end": 326.96, "start": 326.88, "text": "that"}, {"end": 327.48, "start": 326.96, "text": "angle,"}, {"end": 327.76, "start": 327.48, "text": "then"}, {"end": 328.2, "start": 327.76, "text": "this"}, {"end": 328.56, "start": 328.2, "text": "green"}, {"end": 328.96, "start": 328.56, "text": "side"}, {"end": 329.0, "start": 328.96, "text": "would"}, {"end": 329.28, "start": 329.0, "text": "be"}, {"end": 329.32, "start": 329.28, "text": "opposite,"}, {"end": 329.36, "start": 329.32, "text": "and"}, {"end": 329.96, "start": 329.36, "text": "this"}], "text": " We already figured out that the 3 was the opposite side. This is the opposite side. And only when we're talking about this angle. When you talk about this angle, this side is opposite to it. When you talk about this angle, this 4 side is adjacent to it. It's one of the sides that kind of make up that kind of form the vertex here. So this right here is an adjacent side. And I want to be very clear. This only applies to this angle. If we're talking about that angle, then this green side would be opposite, and this"}, {"chunks": [{"end": 330.52, "start": 330.0, "text": "The"}, {"end": 330.6, "start": 330.52, "text": "yellow"}, {"end": 330.6, "start": 330.6, "text": "side"}, {"end": 330.6, "start": 330.6, "text": "would"}, {"end": 330.64, "start": 330.6, "text": "be"}, {"end": 331.28, "start": 330.64, "text": "adjacent."}, {"end": 331.28, "start": 331.28, "text": "But"}, {"end": 331.56, "start": 331.28, "text": "we're"}, {"end": 332.04, "start": 331.56, "text": "just"}, {"end": 332.44, "start": 332.04, "text": "focusing"}, {"end": 332.44, "start": 332.44, "text": "on"}, {"end": 332.88, "start": 332.44, "text": "this"}, {"end": 333.04, "start": 332.88, "text": "angle"}, {"end": 333.64, "start": 333.04, "text": "right"}, {"end": 333.88, "start": 333.64, "text": "over"}, {"end": 334.2, "start": 333.88, "text": "here."}, {"end": 334.36, "start": 334.2, "text": "So"}, {"end": 335.12, "start": 334.36, "text": "cosine"}, {"end": 335.4, "start": 335.12, "text": "of"}, {"end": 335.92, "start": 335.4, "text": "this"}, {"end": 336.08, "start": 335.92, "text": "angle,"}, {"end": 336.32, "start": 336.08, "text": "we"}, {"end": 336.52, "start": 336.32, "text": "hear"}, {"end": 336.8, "start": 336.52, "text": "about"}, {"end": 337.36, "start": 336.8, "text": "adjacent."}, {"end": 337.48, "start": 337.36, "text": "Well,"}, {"end": 337.64, "start": 337.48, "text": "the"}, {"end": 338.04, "start": 337.64, "text": "adjacent"}, {"end": 338.76, "start": 338.04, "text": "side"}, {"end": 339.12, "start": 338.76, "text": "of"}, {"end": 339.2, "start": 339.12, "text": "this"}, {"end": 339.68, "start": 339.2, "text": "angle"}, {"end": 339.96, "start": 339.68, "text": "is"}, {"end": 340.32, "start": 339.96, "text": "4."}, {"end": 340.6, "start": 340.32, "text": "So"}, {"end": 340.6, "start": 340.6, "text": "it"}, {"end": 340.6, "start": 340.6, "text": "is"}, {"end": 341.6, "start": 340.6, "text": "adjacent"}, {"end": 341.92, "start": 341.6, "text": "over"}, {"end": 342.24, "start": 341.92, "text": "the"}, {"end": 342.88, "start": 342.24, "text": "hypotenuse."}, {"end": 343.04, "start": 342.88, "text": "It's"}, {"end": 343.4, "start": 343.04, "text": "the"}, {"end": 344.0, "start": 343.4, "text": "adjacent,"}, {"end": 344.12, "start": 344.0, "text": "which"}, {"end": 344.52, "start": 344.12, "text": "is"}, {"end": 345.16, "start": 344.52, "text": "4,"}, {"end": 345.52, "start": 345.16, "text": "over"}, {"end": 346.0, "start": 345.52, "text": "the"}, {"end": 347.32, "start": 346.0, "text": "hypotenuse."}, {"end": 348.04, "start": 347.32, "text": "4"}, {"end": 349.28, "start": 348.04, "text": "over"}, {"end": 349.8, "start": 349.28, "text": "5."}, {"end": 350.56, "start": 349.8, "text": "Now"}, {"end": 350.92, "start": 350.56, "text": "let's"}, {"end": 351.24, "start": 350.92, "text": "do"}, {"end": 351.24, "start": 351.24, "text": "the"}, {"end": 353.48, "start": 351.24, "text": "tangent."}, {"end": 353.92, "start": 353.48, "text": "Let's"}, {"end": 354.2, "start": 353.92, "text": "do"}, {"end": 354.44, "start": 354.2, "text": "the"}, {"end": 355.84, "start": 354.44, "text": "tangent."}, {"end": 356.0, "start": 355.84, "text": "The"}, {"end": 356.92, "start": 356.0, "text": "tangent"}, {"end": 357.16, "start": 356.92, "text": "of"}, {"end": 357.64, "start": 357.16, "text": "theta."}, {"end": 357.96, "start": 357.64, "text": "Opposite"}, {"end": 358.68, "start": 357.96, "text": "over"}, {"end": 359.96, "start": 358.68, "text": "adjacent."}], "text": " The yellow side would be adjacent. But we're just focusing on this angle right over here. So cosine of this angle, we hear about adjacent. Well, the adjacent side of this angle is 4. So it is adjacent over the hypotenuse. It's the adjacent, which is 4, over the hypotenuse. 4 over 5. Now let's do the tangent. Let's do the tangent. The tangent of theta. Opposite over adjacent."}, {"chunks": [{"end": 360.4, "start": 360.0, "text": "The"}, {"end": 361.04, "start": 360.4, "text": "opposite"}, {"end": 361.72, "start": 361.04, "text": "side"}, {"end": 361.72, "start": 361.72, "text": "is"}, {"end": 362.28, "start": 361.72, "text": "3."}, {"end": 362.68, "start": 362.28, "text": "What"}, {"end": 362.68, "start": 362.68, "text": "is"}, {"end": 362.72, "start": 362.68, "text": "the"}, {"end": 362.96, "start": 362.72, "text": "adjacent"}, {"end": 363.72, "start": 362.96, "text": "side?"}, {"end": 364.44, "start": 363.72, "text": "We"}, {"end": 364.76, "start": 364.44, "text": "already"}, {"end": 366.04, "start": 364.76, "text": "figured"}, {"end": 366.64, "start": 366.04, "text": "that"}, {"end": 366.84, "start": 366.64, "text": "out."}, {"end": 367.0, "start": 366.84, "text": "The"}, {"end": 367.44, "start": 367.0, "text": "adjacent"}, {"end": 368.04, "start": 367.44, "text": "side"}, {"end": 368.44, "start": 368.04, "text": "is"}, {"end": 369.08, "start": 368.44, "text": "4."}, {"end": 369.16, "start": 369.08, "text": "So"}, {"end": 369.52, "start": 369.16, "text": "knowing"}, {"end": 370.16, "start": 369.52, "text": "the"}, {"end": 370.64, "start": 370.16, "text": "sides"}, {"end": 370.68, "start": 370.64, "text": "of"}, {"end": 370.92, "start": 370.68, "text": "this"}, {"end": 371.16, "start": 370.92, "text": "right"}, {"end": 371.56, "start": 371.16, "text": "triangle,"}, {"end": 371.6, "start": 371.56, "text": "we"}, {"end": 371.8, "start": 371.6, "text": "were"}, {"end": 372.16, "start": 371.8, "text": "able"}, {"end": 372.8, "start": 372.16, "text": "to"}, {"end": 373.16, "start": 372.8, "text": "figure"}, {"end": 373.28, "start": 373.16, "text": "out"}, {"end": 373.48, "start": 373.28, "text": "the"}, {"end": 374.0, "start": 373.48, "text": "major"}, {"end": 374.36, "start": 374.0, "text": "trig"}, {"end": 374.8, "start": 374.36, "text": "ratios."}, {"end": 374.8, "start": 374.8, "text": "And"}, {"end": 374.92, "start": 374.8, "text": "we'll"}, {"end": 375.08, "start": 374.92, "text": "see"}, {"end": 375.44, "start": 375.08, "text": "there"}, {"end": 375.6, "start": 375.44, "text": "are"}, {"end": 376.08, "start": 375.6, "text": "other"}, {"end": 376.28, "start": 376.08, "text": "trig"}, {"end": 376.64, "start": 376.28, "text": "ratios,"}, {"end": 376.72, "start": 376.64, "text": "but"}, {"end": 376.84, "start": 376.72, "text": "they"}, {"end": 377.4, "start": 376.84, "text": "can"}, {"end": 377.88, "start": 377.4, "text": "all"}, {"end": 378.12, "start": 377.88, "text": "be"}, {"end": 378.96, "start": 378.12, "text": "derived"}, {"end": 379.28, "start": 378.96, "text": "from"}, {"end": 379.76, "start": 379.28, "text": "these"}, {"end": 380.32, "start": 379.76, "text": "three"}, {"end": 380.92, "start": 380.32, "text": "basic"}, {"end": 381.44, "start": 380.92, "text": "trig"}, {"end": 381.88, "start": 381.44, "text": "functions."}, {"end": 381.88, "start": 381.88, "text": "Now"}, {"end": 382.28, "start": 381.88, "text": "let's"}, {"end": 382.76, "start": 382.28, "text": "think"}, {"end": 383.4, "start": 382.76, "text": "about"}, {"end": 383.72, "start": 383.4, "text": "another"}, {"end": 384.44, "start": 383.72, "text": "angle"}, {"end": 384.48, "start": 384.44, "text": "in"}, {"end": 384.6, "start": 384.48, "text": "this"}, {"end": 384.96, "start": 384.6, "text": "triangle."}, {"end": 385.0, "start": 384.96, "text": "And"}, {"end": 385.28, "start": 385.0, "text": "I'll"}, {"end": 385.92, "start": 385.28, "text": "redraw"}, {"end": 386.12, "start": 385.92, "text": "it,"}, {"end": 386.2, "start": 386.12, "text": "just"}, {"end": 386.44, "start": 386.2, "text": "because"}, {"end": 386.6, "start": 386.44, "text": "my"}, {"end": 387.76, "start": 386.6, "text": "triangle"}, {"end": 387.88, "start": 387.76, "text": "is"}, {"end": 388.64, "start": 387.88, "text": "getting"}, {"end": 388.96, "start": 388.64, "text": "a"}, {"end": 389.48, "start": 388.96, "text": "little"}, {"end": 389.6, "start": 389.48, "text": "bit"}, {"end": 389.96, "start": 389.6, "text": "messy."}], "text": " The opposite side is 3. What is the adjacent side? We already figured that out. The adjacent side is 4. So knowing the sides of this right triangle, we were able to figure out the major trig ratios. And we'll see there are other trig ratios, but they can all be derived from these three basic trig functions. Now let's think about another angle in this triangle. And I'll redraw it, just because my triangle is getting a little bit messy."}, {"chunks": [{"end": 390.52, "start": 390.0, "text": "And"}, {"end": 391.8, "start": 390.52, "text": "once"}, {"end": 392.08, "start": 391.8, "text": "again,"}, {"end": 392.52, "start": 392.08, "text": "the"}, {"end": 393.12, "start": 392.52, "text": "lengths"}, {"end": 393.68, "start": 393.12, "text": "of"}, {"end": 394.12, "start": 393.68, "text": "this"}, {"end": 394.88, "start": 394.12, "text": "triangle"}, {"end": 395.0, "start": 394.88, "text": "are,"}, {"end": 395.32, "start": 395.0, "text": "we"}, {"end": 395.48, "start": 395.32, "text": "have"}, {"end": 395.88, "start": 395.48, "text": "length"}, {"end": 397.44, "start": 395.88, "text": "4"}, {"end": 397.72, "start": 397.44, "text": "there,"}, {"end": 398.16, "start": 397.72, "text": "we"}, {"end": 398.64, "start": 398.16, "text": "have"}, {"end": 399.08, "start": 398.64, "text": "length"}, {"end": 399.44, "start": 399.08, "text": "3"}, {"end": 400.0, "start": 399.44, "text": "there,"}, {"end": 400.56, "start": 400.0, "text": "and"}, {"end": 401.04, "start": 400.56, "text": "we"}, {"end": 401.64, "start": 401.04, "text": "have"}, {"end": 402.08, "start": 401.64, "text": "length"}, {"end": 402.52, "start": 402.08, "text": "5"}, {"end": 403.12, "start": 402.52, "text": "there."}, {"end": 403.36, "start": 403.12, "text": "The"}, {"end": 404.04, "start": 403.36, "text": "last"}, {"end": 404.32, "start": 404.04, "text": "example,"}, {"end": 404.96, "start": 404.32, "text": "we"}, {"end": 405.2, "start": 404.96, "text": "used"}, {"end": 405.68, "start": 405.2, "text": "this"}, {"end": 406.4, "start": 405.68, "text": "theta."}, {"end": 406.72, "start": 406.4, "text": "But"}, {"end": 407.68, "start": 406.72, "text": "let's"}, {"end": 408.16, "start": 407.68, "text": "do"}, {"end": 408.6, "start": 408.16, "text": "another"}, {"end": 409.36, "start": 408.6, "text": "angle"}, {"end": 409.56, "start": 409.36, "text": "up"}, {"end": 409.88, "start": 409.56, "text": "here."}, {"end": 410.08, "start": 409.88, "text": "And"}, {"end": 410.48, "start": 410.08, "text": "let's"}, {"end": 410.68, "start": 410.48, "text": "call"}, {"end": 412.28, "start": 410.68, "text": "this"}, {"end": 412.64, "start": 412.28, "text": "angle,"}, {"end": 413.04, "start": 412.64, "text": "I"}, {"end": 413.4, "start": 413.04, "text": "don't"}, {"end": 413.8, "start": 413.4, "text": "know,"}, {"end": 414.04, "start": 413.8, "text": "I'll"}, {"end": 414.8, "start": 414.04, "text": "think"}, {"end": 415.44, "start": 414.8, "text": "of"}, {"end": 415.88, "start": 415.44, "text": "something,"}, {"end": 416.44, "start": 415.88, "text": "a"}, {"end": 416.72, "start": 416.44, "text": "random"}, {"end": 417.0, "start": 416.72, "text": "Greek"}, {"end": 417.32, "start": 417.0, "text": "letter."}, {"end": 417.6, "start": 417.32, "text": "So"}, {"end": 418.12, "start": 417.6, "text": "let's"}, {"end": 418.4, "start": 418.12, "text": "say"}, {"end": 418.6, "start": 418.4, "text": "it's"}, {"end": 419.96, "start": 418.6, "text": "zai."}], "text": " And once again, the lengths of this triangle are, we have length 4 there, we have length 3 there, and we have length 5 there. The last example, we used this theta. But let's do another angle up here. And let's call this angle, I don't know, I'll think of something, a random Greek letter. So let's say it's zai."}, {"chunks": [{"end": 420.28, "start": 420.0, "text": "It's"}, {"end": 420.36, "start": 420.28, "text": "a"}, {"end": 420.48, "start": 420.36, "text": "little"}, {"end": 420.56, "start": 420.48, "text": "bit"}, {"end": 420.96, "start": 420.56, "text": "bizarre."}, {"end": 421.12, "start": 420.96, "text": "Theta"}, {"end": 421.12, "start": 421.12, "text": "is"}, {"end": 421.16, "start": 421.12, "text": "what"}, {"end": 421.72, "start": 421.16, "text": "you"}, {"end": 422.24, "start": 421.72, "text": "normally"}, {"end": 422.48, "start": 422.24, "text": "use,"}, {"end": 422.8, "start": 422.48, "text": "but"}, {"end": 423.44, "start": 422.8, "text": "since"}, {"end": 423.72, "start": 423.44, "text": "I"}, {"end": 424.0, "start": 423.72, "text": "already"}, {"end": 424.76, "start": 424.0, "text": "used"}, {"end": 425.44, "start": 424.76, "text": "theta,"}, {"end": 425.88, "start": 425.44, "text": "let's"}, {"end": 426.36, "start": 425.88, "text": "use"}, {"end": 426.76, "start": 426.36, "text": "xi."}, {"end": 426.92, "start": 426.76, "text": "Actually,"}, {"end": 427.36, "start": 426.92, "text": "instead"}, {"end": 427.52, "start": 427.36, "text": "of"}, {"end": 427.52, "start": 427.52, "text": "xi,"}, {"end": 427.52, "start": 427.52, "text": "let"}, {"end": 427.72, "start": 427.52, "text": "me"}, {"end": 427.96, "start": 427.72, "text": "just"}, {"end": 428.6, "start": 427.96, "text": "simplify"}, {"end": 428.6, "start": 428.6, "text": "it."}, {"end": 428.6, "start": 428.6, "text": "Let"}, {"end": 428.6, "start": 428.6, "text": "me"}, {"end": 428.64, "start": 428.6, "text": "call"}, {"end": 428.64, "start": 428.64, "text": "this"}, {"end": 428.68, "start": 428.64, "text": "angle"}, {"end": 428.92, "start": 428.68, "text": "x."}, {"end": 429.48, "start": 428.92, "text": "Let's"}, {"end": 429.64, "start": 429.48, "text": "call"}, {"end": 429.64, "start": 429.64, "text": "that"}, {"end": 429.76, "start": 429.64, "text": "angle"}, {"end": 430.08, "start": 429.76, "text": "x."}, {"end": 430.44, "start": 430.08, "text": "So"}, {"end": 430.44, "start": 430.44, "text": "let's"}, {"end": 430.44, "start": 430.44, "text": "figure"}, {"end": 430.44, "start": 430.44, "text": "out"}, {"end": 430.52, "start": 430.44, "text": "the"}, {"end": 430.8, "start": 430.52, "text": "trig"}, {"end": 431.32, "start": 430.8, "text": "functions"}, {"end": 431.68, "start": 431.32, "text": "for"}, {"end": 432.0, "start": 431.68, "text": "that"}, {"end": 432.56, "start": 432.0, "text": "angle"}, {"end": 432.84, "start": 432.56, "text": "x."}, {"end": 433.0, "start": 432.84, "text": "So"}, {"end": 433.32, "start": 433.0, "text": "we"}, {"end": 433.48, "start": 433.32, "text": "have"}, {"end": 435.16, "start": 433.48, "text": "sine"}, {"end": 435.36, "start": 435.16, "text": "of"}, {"end": 435.76, "start": 435.36, "text": "x"}, {"end": 436.08, "start": 435.76, "text": "is"}, {"end": 436.16, "start": 436.08, "text": "going"}, {"end": 436.16, "start": 436.16, "text": "to"}, {"end": 436.44, "start": 436.16, "text": "be"}, {"end": 436.6, "start": 436.44, "text": "equal"}, {"end": 436.64, "start": 436.6, "text": "to"}, {"end": 437.8, "start": 436.64, "text": "what?"}, {"end": 438.04, "start": 437.8, "text": "Well,"}, {"end": 438.52, "start": 438.04, "text": "sine"}, {"end": 438.88, "start": 438.52, "text": "is"}, {"end": 439.48, "start": 438.88, "text": "opposite"}, {"end": 439.72, "start": 439.48, "text": "over"}, {"end": 440.4, "start": 439.72, "text": "hypotenuse."}, {"end": 440.64, "start": 440.4, "text": "So"}, {"end": 441.0, "start": 440.64, "text": "what"}, {"end": 441.64, "start": 441.0, "text": "side"}, {"end": 441.76, "start": 441.64, "text": "is"}, {"end": 442.36, "start": 441.76, "text": "opposite"}, {"end": 443.0, "start": 442.36, "text": "to"}, {"end": 443.2, "start": 443.0, "text": "x?"}, {"end": 443.2, "start": 443.2, "text": "Well,"}, {"end": 443.24, "start": 443.2, "text": "it"}, {"end": 443.64, "start": 443.24, "text": "opens"}, {"end": 443.64, "start": 443.64, "text": "onto"}, {"end": 443.68, "start": 443.64, "text": "this"}, {"end": 444.04, "start": 443.68, "text": "4."}, {"end": 444.04, "start": 444.04, "text": "It"}, {"end": 444.08, "start": 444.04, "text": "opens"}, {"end": 444.28, "start": 444.08, "text": "onto"}, {"end": 444.76, "start": 444.28, "text": "the"}, {"end": 445.8, "start": 444.76, "text": "4."}, {"end": 446.12, "start": 445.8, "text": "So"}, {"end": 446.12, "start": 446.12, "text": "in"}, {"end": 446.6, "start": 446.12, "text": "this"}, {"end": 447.84, "start": 446.6, "text": "context,"}, {"end": 448.2, "start": 447.84, "text": "this"}, {"end": 448.88, "start": 448.2, "text": "is"}, {"end": 449.12, "start": 448.88, "text": "now"}, {"end": 449.12, "start": 449.12, "text": "the"}, {"end": 449.96, "start": 449.12, "text": "opposite."}], "text": " It's a little bit bizarre. Theta is what you normally use, but since I already used theta, let's use xi. Actually, instead of xi, let me just simplify it. Let me call this angle x. Let's call that angle x. So let's figure out the trig functions for that angle x. So we have sine of x is going to be equal to what? Well, sine is opposite over hypotenuse. So what side is opposite to x? Well, it opens onto this 4. It opens onto the 4. So in this context, this is now the opposite."}, {"chunks": [{"end": 450.32, "start": 450.0, "text": "This"}, {"end": 450.88, "start": 450.32, "text": "is"}, {"end": 451.52, "start": 450.88, "text": "now"}, {"end": 451.8, "start": 451.52, "text": "the"}, {"end": 452.04, "start": 451.8, "text": "opposite"}, {"end": 452.28, "start": 452.04, "text": "side."}, {"end": 452.88, "start": 452.28, "text": "Remember,"}, {"end": 453.24, "start": 452.88, "text": "4"}, {"end": 453.48, "start": 453.24, "text": "was"}, {"end": 453.96, "start": 453.48, "text": "adjacent"}, {"end": 454.12, "start": 453.96, "text": "to"}, {"end": 454.44, "start": 454.12, "text": "this"}, {"end": 454.88, "start": 454.44, "text": "theta,"}, {"end": 455.16, "start": 454.88, "text": "but"}, {"end": 455.64, "start": 455.16, "text": "it's"}, {"end": 456.24, "start": 455.64, "text": "opposite"}, {"end": 456.48, "start": 456.24, "text": "to"}, {"end": 457.0, "start": 456.48, "text": "x."}, {"end": 457.12, "start": 457.0, "text": "So"}, {"end": 457.32, "start": 457.12, "text": "it's"}, {"end": 457.4, "start": 457.32, "text": "going"}, {"end": 457.8, "start": 457.4, "text": "to"}, {"end": 458.0, "start": 457.8, "text": "be"}, {"end": 459.0, "start": 458.0, "text": "4"}, {"end": 459.6, "start": 459.0, "text": "over,"}, {"end": 459.76, "start": 459.6, "text": "now"}, {"end": 459.96, "start": 459.76, "text": "what's"}, {"end": 460.24, "start": 459.96, "text": "the"}, {"end": 460.68, "start": 460.24, "text": "hypotenuse?"}, {"end": 460.8, "start": 460.68, "text": "Well,"}, {"end": 461.04, "start": 460.8, "text": "the"}, {"end": 461.68, "start": 461.04, "text": "hypotenuse"}, {"end": 461.68, "start": 461.68, "text": "is"}, {"end": 461.68, "start": 461.68, "text": "going"}, {"end": 461.68, "start": 461.68, "text": "to"}, {"end": 461.68, "start": 461.68, "text": "be"}, {"end": 461.88, "start": 461.68, "text": "the"}, {"end": 462.32, "start": 461.88, "text": "same"}, {"end": 462.72, "start": 462.32, "text": "regardless"}, {"end": 462.76, "start": 462.72, "text": "of"}, {"end": 463.12, "start": 462.76, "text": "which"}, {"end": 463.48, "start": 463.12, "text": "angle"}, {"end": 463.8, "start": 463.48, "text": "you"}, {"end": 464.2, "start": 463.8, "text": "pick."}, {"end": 464.32, "start": 464.2, "text": "So"}, {"end": 464.56, "start": 464.32, "text": "the"}, {"end": 465.08, "start": 464.56, "text": "hypotenuse"}, {"end": 465.28, "start": 465.08, "text": "is"}, {"end": 465.64, "start": 465.28, "text": "now"}, {"end": 465.76, "start": 465.64, "text": "going"}, {"end": 466.0, "start": 465.76, "text": "to"}, {"end": 466.12, "start": 466.0, "text": "be"}, {"end": 466.76, "start": 466.12, "text": "5."}, {"end": 466.88, "start": 466.76, "text": "So"}, {"end": 467.12, "start": 466.88, "text": "it's"}, {"end": 467.32, "start": 467.12, "text": "4"}, {"end": 467.92, "start": 467.32, "text": "fifths."}, {"end": 468.08, "start": 467.92, "text": "Now"}, {"end": 468.16, "start": 468.08, "text": "let's"}, {"end": 468.52, "start": 468.16, "text": "do"}, {"end": 469.16, "start": 468.52, "text": "another"}, {"end": 469.48, "start": 469.16, "text": "one."}, {"end": 469.72, "start": 469.48, "text": "What"}, {"end": 470.64, "start": 469.72, "text": "is"}, {"end": 470.88, "start": 470.64, "text": "the"}, {"end": 472.16, "start": 470.88, "text": "cosine?"}, {"end": 472.4, "start": 472.16, "text": "What"}, {"end": 472.8, "start": 472.4, "text": "is"}, {"end": 472.84, "start": 472.8, "text": "the"}, {"end": 473.48, "start": 472.84, "text": "cosine"}, {"end": 473.72, "start": 473.48, "text": "of"}, {"end": 474.36, "start": 473.72, "text": "x?"}, {"end": 474.68, "start": 474.36, "text": "So"}, {"end": 475.84, "start": 474.68, "text": "cosine"}, {"end": 475.84, "start": 475.84, "text": "is"}, {"end": 476.56, "start": 475.84, "text": "adjacent"}, {"end": 476.76, "start": 476.56, "text": "over"}, {"end": 477.32, "start": 476.76, "text": "hypotenuse."}, {"end": 477.44, "start": 477.32, "text": "What"}, {"end": 478.08, "start": 477.44, "text": "side"}, {"end": 478.44, "start": 478.08, "text": "is"}, {"end": 479.24, "start": 478.44, "text": "adjacent"}, {"end": 479.28, "start": 479.24, "text": "to"}, {"end": 479.96, "start": 479.28, "text": "x?"}], "text": " This is now the opposite side. Remember, 4 was adjacent to this theta, but it's opposite to x. So it's going to be 4 over, now what's the hypotenuse? Well, the hypotenuse is going to be the same regardless of which angle you pick. So the hypotenuse is now going to be 5. So it's 4 fifths. Now let's do another one. What is the cosine? What is the cosine of x? So cosine is adjacent over hypotenuse. What side is adjacent to x?"}, {"chunks": [{"end": 480.52, "start": 480.0, "text": "You"}, {"end": 480.64, "start": 480.52, "text": "have"}, {"end": 480.68, "start": 480.64, "text": "the"}, {"end": 481.12, "start": 480.68, "text": "hypotenuse"}, {"end": 481.84, "start": 481.12, "text": "here."}, {"end": 481.92, "start": 481.84, "text": "Well,"}, {"end": 482.12, "start": 481.92, "text": "the"}, {"end": 482.4, "start": 482.12, "text": "3"}, {"end": 482.84, "start": 482.4, "text": "side,"}, {"end": 483.12, "start": 482.84, "text": "it's"}, {"end": 483.48, "start": 483.12, "text": "one"}, {"end": 484.04, "start": 483.48, "text": "of"}, {"end": 484.4, "start": 484.04, "text": "the"}, {"end": 484.8, "start": 484.4, "text": "sides"}, {"end": 484.84, "start": 484.8, "text": "that"}, {"end": 485.44, "start": 484.84, "text": "forms"}, {"end": 485.6, "start": 485.44, "text": "the"}, {"end": 486.04, "start": 485.6, "text": "vertex"}, {"end": 486.08, "start": 486.04, "text": "that"}, {"end": 486.64, "start": 486.08, "text": "x"}, {"end": 486.96, "start": 486.64, "text": "is"}, {"end": 487.24, "start": 486.96, "text": "at,"}, {"end": 487.36, "start": 487.24, "text": "and"}, {"end": 487.36, "start": 487.36, "text": "it's"}, {"end": 487.36, "start": 487.36, "text": "not"}, {"end": 487.4, "start": 487.36, "text": "the"}, {"end": 487.76, "start": 487.4, "text": "hypotenuse,"}, {"end": 488.0, "start": 487.76, "text": "so"}, {"end": 488.44, "start": 488.0, "text": "this"}, {"end": 488.52, "start": 488.44, "text": "is"}, {"end": 488.52, "start": 488.52, "text": "the"}, {"end": 488.96, "start": 488.52, "text": "adjacent"}, {"end": 489.28, "start": 488.96, "text": "side."}, {"end": 489.6, "start": 489.28, "text": "That"}, {"end": 490.04, "start": 489.6, "text": "is"}, {"end": 490.76, "start": 490.04, "text": "adjacent,"}, {"end": 490.84, "start": 490.76, "text": "so"}, {"end": 491.04, "start": 490.84, "text": "it's"}, {"end": 491.2, "start": 491.04, "text": "equal"}, {"end": 491.52, "start": 491.2, "text": "to"}, {"end": 492.48, "start": 491.52, "text": "3"}, {"end": 492.84, "start": 492.48, "text": "over"}, {"end": 493.24, "start": 492.84, "text": "the"}, {"end": 494.04, "start": 493.24, "text": "hypotenuse."}, {"end": 494.08, "start": 494.04, "text": "The"}, {"end": 495.12, "start": 494.08, "text": "hypotenuse"}, {"end": 496.32, "start": 495.12, "text": "is"}, {"end": 496.44, "start": 496.32, "text": "5."}, {"end": 496.48, "start": 496.44, "text": "And"}, {"end": 496.8, "start": 496.48, "text": "then"}, {"end": 497.6, "start": 496.8, "text": "finally,"}, {"end": 497.72, "start": 497.6, "text": "the"}, {"end": 499.64, "start": 497.72, "text": "tangent."}, {"end": 499.88, "start": 499.64, "text": "We"}, {"end": 500.48, "start": 499.88, "text": "want"}, {"end": 500.64, "start": 500.48, "text": "to"}, {"end": 501.08, "start": 500.64, "text": "figure"}, {"end": 501.64, "start": 501.08, "text": "out"}, {"end": 501.88, "start": 501.64, "text": "the"}, {"end": 501.92, "start": 501.88, "text": "tangent"}, {"end": 502.12, "start": 501.92, "text": "of"}, {"end": 502.2, "start": 502.12, "text": "x."}, {"end": 502.68, "start": 502.2, "text": "Tangent"}, {"end": 503.12, "start": 502.68, "text": "is"}, {"end": 503.52, "start": 503.12, "text": "opposite"}, {"end": 503.8, "start": 503.52, "text": "over"}, {"end": 504.2, "start": 503.8, "text": "adjacent."}, {"end": 504.64, "start": 504.2, "text": "Soh"}, {"end": 504.92, "start": 504.64, "text": "ka"}, {"end": 505.36, "start": 504.92, "text": "toa."}, {"end": 505.84, "start": 505.36, "text": "Tangent"}, {"end": 506.52, "start": 505.84, "text": "is"}, {"end": 507.4, "start": 506.52, "text": "opposite"}, {"end": 507.88, "start": 507.4, "text": "over"}, {"end": 508.44, "start": 507.88, "text": "adjacent."}, {"end": 509.28, "start": 508.44, "text": "Opposite"}, {"end": 509.48, "start": 509.28, "text": "over"}, {"end": 509.96, "start": 509.48, "text": "adjacent."}], "text": " You have the hypotenuse here. Well, the 3 side, it's one of the sides that forms the vertex that x is at, and it's not the hypotenuse, so this is the adjacent side. That is adjacent, so it's equal to 3 over the hypotenuse. The hypotenuse is 5. And then finally, the tangent. We want to figure out the tangent of x. Tangent is opposite over adjacent. Soh ka toa. Tangent is opposite over adjacent. Opposite over adjacent."}, {"chunks": [{"end": 510.36, "start": 510.0, "text": "The"}, {"end": 511.16, "start": 510.36, "text": "opposite"}, {"end": 511.92, "start": 511.16, "text": "side"}, {"end": 512.04, "start": 511.92, "text": "is"}, {"end": 512.44, "start": 512.04, "text": "4."}, {"end": 512.72, "start": 512.44, "text": "I"}, {"end": 512.96, "start": 512.72, "text": "want"}, {"end": 513.12, "start": 512.96, "text": "to"}, {"end": 513.16, "start": 513.12, "text": "do"}, {"end": 513.2, "start": 513.16, "text": "it"}, {"end": 513.48, "start": 513.2, "text": "in"}, {"end": 514.0, "start": 513.48, "text": "that"}, {"end": 514.04, "start": 514.0, "text": "blue"}, {"end": 514.68, "start": 514.04, "text": "color."}, {"end": 514.96, "start": 514.68, "text": "The"}, {"end": 515.4, "start": 514.96, "text": "opposite"}, {"end": 516.0, "start": 515.4, "text": "side"}, {"end": 516.32, "start": 516.0, "text": "is"}, {"end": 516.96, "start": 516.32, "text": "4."}, {"end": 517.24, "start": 516.96, "text": "And"}, {"end": 517.6, "start": 517.24, "text": "the"}, {"end": 518.52, "start": 517.6, "text": "adjacent"}, {"end": 519.56, "start": 518.52, "text": "side"}, {"end": 519.68, "start": 519.56, "text": "is"}, {"end": 519.96, "start": 519.68, "text": "3."}, {"end": 519.96, "start": 519.96, "text": "And"}, {"end": 520.52, "start": 519.96, "text": "we're"}, {"end": 521.12, "start": 520.52, "text": "done."}, {"end": 521.16, "start": 521.12, "text": "In"}, {"end": 521.68, "start": 521.16, "text": "the"}, {"end": 522.44, "start": 521.68, "text": "next"}, {"end": 523.16, "start": 522.44, "text": "video,"}, {"end": 523.44, "start": 523.16, "text": "I'll"}, {"end": 523.48, "start": 523.44, "text": "do"}, {"end": 523.72, "start": 523.48, "text": "a"}, {"end": 524.56, "start": 523.72, "text": "ton"}, {"end": 524.72, "start": 524.56, "text": "of"}, {"end": 524.88, "start": 524.72, "text": "more"}, {"end": 525.68, "start": 524.88, "text": "examples"}, {"end": 525.72, "start": 525.68, "text": "of"}, {"end": 525.76, "start": 525.72, "text": "this,"}, {"end": 526.12, "start": 525.76, "text": "just"}, {"end": 526.4, "start": 526.12, "text": "so"}, {"end": 526.56, "start": 526.4, "text": "we"}, {"end": 526.88, "start": 526.56, "text": "really"}, {"end": 526.88, "start": 526.88, "text": "get"}, {"end": 526.88, "start": 526.88, "text": "a"}, {"end": 526.88, "start": 526.88, "text": "feel"}, {"end": 526.88, "start": 526.88, "text": "for"}, {"end": 526.88, "start": 526.88, "text": "it."}, {"end": 526.88, "start": 526.88, "text": "But"}, {"end": 526.88, "start": 526.88, "text": "I'll"}, {"end": 526.88, "start": 526.88, "text": "leave"}, {"end": 526.88, "start": 526.88, "text": "you"}, {"end": 526.88, "start": 526.88, "text": "thinking"}, {"end": 526.88, "start": 526.88, "text": "of"}, {"end": 526.88, "start": 526.88, "text": "what"}, {"end": 526.88, "start": 526.88, "text": "happens"}, {"end": 526.88, "start": 526.88, "text": "when"}, {"end": 527.44, "start": 526.88, "text": "these"}, {"end": 528.2, "start": 527.44, "text": "angles"}, {"end": 529.04, "start": 528.2, "text": "start"}, {"end": 529.44, "start": 529.04, "text": "to"}, {"end": 530.04, "start": 529.44, "text": "approach"}, {"end": 530.6, "start": 530.04, "text": "90"}, {"end": 531.28, "start": 530.6, "text": "degrees,"}, {"end": 531.44, "start": 531.28, "text": "or"}, {"end": 531.6, "start": 531.44, "text": "how"}, {"end": 531.92, "start": 531.6, "text": "could"}, {"end": 532.2, "start": 531.92, "text": "they"}, {"end": 532.36, "start": 532.2, "text": "even"}, {"end": 532.72, "start": 532.36, "text": "get"}, {"end": 533.0, "start": 532.72, "text": "larger"}, {"end": 533.0, "start": 533.0, "text": "than"}, {"end": 533.32, "start": 533.0, "text": "90"}, {"end": 533.48, "start": 533.32, "text": "degrees?"}, {"end": 533.48, "start": 533.48, "text": "And"}, {"end": 533.48, "start": 533.48, "text": "what"}, {"end": 533.6, "start": 533.48, "text": "we're"}, {"end": 534.24, "start": 533.6, "text": "going"}, {"end": 534.4, "start": 534.24, "text": "to"}, {"end": 534.6, "start": 534.4, "text": "see"}, {"end": 535.08, "start": 534.6, "text": "is"}, {"end": 535.48, "start": 535.08, "text": "that"}, {"end": 535.84, "start": 535.48, "text": "this"}, {"end": 536.56, "start": 535.84, "text": "definition,"}, {"end": 536.64, "start": 536.56, "text": "the"}, {"end": 537.0, "start": 536.64, "text": "SOHCAH"}, {"end": 537.28, "start": 537.0, "text": "TOA"}, {"end": 537.72, "start": 537.28, "text": "definition,"}, {"end": 538.28, "start": 537.72, "text": "takes"}, {"end": 538.56, "start": 538.28, "text": "us"}, {"end": 539.04, "start": 538.56, "text": "a"}, {"end": 539.12, "start": 539.04, "text": "long"}, {"end": 539.64, "start": 539.12, "text": "way"}, {"end": 539.92, "start": 539.64, "text": "for"}, {"end": 539.92, "start": 539.92, "text": "angles"}, {"end": 539.92, "start": 539.92, "text": "that"}, {"end": 539.96, "start": 539.92, "text": "are"}], "text": " The opposite side is 4. I want to do it in that blue color. The opposite side is 4. And the adjacent side is 3. And we're done. In the next video, I'll do a ton of more examples of this, just so we really get a feel for it. But I'll leave you thinking of what happens when these angles start to approach 90 degrees, or how could they even get larger than 90 degrees? And what we're going to see is that this definition, the SOHCAH TOA definition, takes us a long way for angles that are"}, {"chunks": [{"end": 540.36, "start": 540.0, "text": "between"}, {"end": 541.0, "start": 540.36, "text": "0"}, {"end": 541.2, "start": 541.0, "text": "and"}, {"end": 541.32, "start": 541.2, "text": "90"}, {"end": 541.68, "start": 541.32, "text": "degrees,"}, {"end": 541.8, "start": 541.68, "text": "or"}, {"end": 542.16, "start": 541.8, "text": "that"}, {"end": 542.32, "start": 542.16, "text": "are"}, {"end": 542.6, "start": 542.32, "text": "less"}, {"end": 542.72, "start": 542.6, "text": "than"}, {"end": 543.0, "start": 542.72, "text": "90"}, {"end": 543.48, "start": 543.0, "text": "degrees."}, {"end": 543.52, "start": 543.48, "text": "But"}, {"end": 543.96, "start": 543.52, "text": "they"}, {"end": 544.32, "start": 543.96, "text": "kind"}, {"end": 544.44, "start": 544.32, "text": "of"}, {"end": 544.8, "start": 544.44, "text": "start"}, {"end": 544.88, "start": 544.8, "text": "to"}, {"end": 545.28, "start": 544.88, "text": "mess"}, {"end": 545.44, "start": 545.28, "text": "up"}, {"end": 545.84, "start": 545.44, "text": "really"}, {"end": 545.96, "start": 545.84, "text": "at"}, {"end": 546.12, "start": 545.96, "text": "the"}, {"end": 546.64, "start": 546.12, "text": "boundaries."}, {"end": 546.68, "start": 546.64, "text": "And"}, {"end": 546.84, "start": 546.68, "text": "we're"}, {"end": 546.84, "start": 546.84, "text": "going"}, {"end": 546.96, "start": 546.84, "text": "to"}, {"end": 547.6, "start": 546.96, "text": "introduce"}, {"end": 547.76, "start": 547.6, "text": "a"}, {"end": 547.8, "start": 547.76, "text": "new"}, {"end": 548.32, "start": 547.8, "text": "definition"}, {"end": 548.76, "start": 548.32, "text": "that's"}, {"end": 549.28, "start": 548.76, "text": "kind"}, {"end": 549.32, "start": 549.28, "text": "of"}, {"end": 549.6, "start": 549.32, "text": "derived"}, {"end": 549.84, "start": 549.6, "text": "from"}, {"end": 549.88, "start": 549.84, "text": "the"}, {"end": 550.16, "start": 549.88, "text": "SOHCAHTOA"}, {"end": 550.68, "start": 550.16, "text": "definition"}, {"end": 550.96, "start": 550.68, "text": "for"}, {"end": 551.48, "start": 550.96, "text": "finding"}, {"end": 551.56, "start": 551.48, "text": "the"}, {"end": 552.04, "start": 551.56, "text": "sine,"}, {"end": 552.68, "start": 552.04, "text": "cosine,"}, {"end": 552.96, "start": 552.68, "text": "and"}, {"end": 553.2, "start": 552.96, "text": "tangent"}, {"end": 553.36, "start": 553.2, "text": "of"}, {"end": 553.96, "start": 553.36, "text": "really"}, {"end": 554.36, "start": 553.96, "text": "any"}, {"end": 556.36, "start": 554.36, "text": "angle."}], "text": " between 0 and 90 degrees, or that are less than 90 degrees. But they kind of start to mess up really at the boundaries. And we're going to introduce a new definition that's kind of derived from the SOHCAHTOA definition for finding the sine, cosine, and tangent of really any angle."}]}}