Truly Understand Trigonometry — Transcript

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1. 0:00In nature, we observe numerous phenomena
2. 0:03that propagate in the form of waves or
3. 0:05can be described by the concept of a
4. 0:07wave. To understand these phenomena,
5. 0:09[music] we model them mathematically.
6. 0:12It is remarkable that the mathematical
7. 0:14description of a wave even when solving
8. 0:16complicated differential equations can
9. 0:18be traced back to a simple geometric
10. 0:20shape, the triangle. So, how [music] can
11. 0:23a wave arise from a triangle? Let's find
12. 0:26out.
13. 0:29>> [music]
14. 0:32>> First of all, there are an infinite
15. 0:34number of triangles with different
16. 0:35sizes, orientations, [music]
17. 0:37and angles. But which one is the
18. 0:39coolest? Well, for the Babylonians, it
19. 0:43was the equilateral triangle. They were
20. 0:45[music] so impressed by it that they
21. 0:47even formulated the concept of an angle
22. 0:49based on it. [music] In a sense, what
23. 0:51they observed between two sides was one
24. 0:54angle.
25. 0:55>> [music]
26. 0:55>> And because all sides of this triangle
27. 0:57are equal in length, it had three of
28. 0:59them.
29. 1:01And [music] because these people used
30. 1:02the sexesimal system for the concept of
31. 1:05numbers, which is a number system based
32. 1:07on the number 60, [music]
33. 1:08they loved to divide different things
34. 1:10into 60 parts. Not only were units of
35. 1:13time divided into 60 parts, [music] but
36. 1:15also units of space. So they imagined
37. 1:18that this angle also consisted of 60
38. 1:21equal parts. [music]
39. 1:23One angle therefore corresponds to 60 um
40. 1:26angle [music] parts which we now call
41. 1:28degrees and symbolize with a circle.
42. 1:32Since exactly six of these form a
43. 1:35complete circle, we measure 6 * 60 or
44. 1:38360° in a circle.
45. 1:42When humans later discovered that they
46. 1:44had 10 fingers and began using the
47. 1:46decimal number system, [music] they were
48. 1:48not very amused by the fact that the
49. 1:50Babylonians divided spacetime by 60.
50. 1:53[music] So they divided the units of
51. 1:56time further into powers of 10 and the
52. 1:59angle measurement also had to be
53. 2:00changed. For this purpose, a fascinating
54. 2:04property of a circle was used. If we
55. 2:06divide the length of the circumference C
56. 2:09by the length of the diameter D, that is
57. 2:12if we calculate the ratio of
58. 2:14circumference to diameter, we get a
59. 2:16number that always remains the same no
60. 2:19matter how large or small we make the
61. 2:21circle. Today, we know this number as
62. 2:24pi.
63. 2:26If we rearrange the equation and set the
64. 2:28diameter equal to 2 * the radius, the
65. 2:31[music] circumference corresponds to 2
66. 2:33pi * the radius. with a radius of 1. The
67. 2:37circumference is simply 2 * [music] pi.
68. 2:39And it was precisely this circle that
69. 2:42was used as a template for a new
70. 2:43measurement of angular units, which is
71. 2:46why it became known as the unit circle.
72. 2:49Instead of imagining that, for example,
73. 2:51[music] a 60° angle consists of 60
74. 2:54parts, we place our circle template with
75. 2:57its center at the tip of the angle and
76. 2:59measure the length of [music] the ark
77. 3:01that this angle encloses.
78. 3:04In the case of this angle, the ark
79. 3:06length corresponds to 16th of the
80. 3:08[music] entire circumference. That is 2
81. 3:11pi / 6 or p<unk>/3 in reduced form.
82. 3:16At a 90° angle, the ark occupies a
83. 3:18quarter of the circle. So this angle
84. 3:21corresponds to a quarter of [music] 2 pi
85. 3:23or half of pi for short. Because we use
86. 3:26this angle so often and want to
87. 3:28recognize it quickly, we mark it with an
88. 3:30additional dot. And for a full circle,
89. 3:33we eventually measured the complete
90. 3:35circumference of 2 pi.
91. 3:39Oh, and the coolest triangle was no
92. 3:41longer the equilateral triangle, but the
93. 3:43right angle triangle because it can do
94. 3:45everything that other triangles can do.
95. 3:47And in addition, the Pythagorean theorem
96. 3:50applies to it. Furthermore, we can
97. 3:52construct any other triangle from two
98. 3:55right angled ones. So if we understand
99. 3:57the right angle triangle, we understand
100. 3:59all [music] the others.
101. 4:01for the study of triangles known as
102. 4:04trigonometry. It is therefore sufficient
103. 4:06to examine the right angle triangle.
104. 4:10All right, let's first name the
105. 4:11individual sides. The side opposite the
106. 4:14right angle is called the hypotenuse.
107. 4:16The other ones are called legs. From the
108. 4:19perspective of an angle, which I will
109. 4:20simply call alpha, we can also name the
110. 4:23legs more precisely. The side opposite
111. 4:25to alpha is then called the opposite
112. 4:27leg. and the one adjacent to the angle
113. 4:29is called the adjacent leg. From the
114. 4:32perspective of the other angle, we would
115. 4:34swap these accordingly.
116. 4:40With these three sides, we can create a
117. 4:42total of six ratios [music] whereby in
118. 4:44the last three ratios only the numerator
119. 4:46and the denominator of the first three
120. 4:48are [music] swapped. And now comes the
121. 4:51remarkable part. As with the circle,
122. 4:54these ratios also correspond to a number
123. 4:56that does not change no matter how large
124. 4:59or small we make the triangle. For the
125. 5:02number to change, for example, in the
126. 5:04first ratio, we would have to change
127. 5:06only the numerator [music] or only the
128. 5:08denominator.
129. 5:10But how can we change the numerator,
130. 5:12which is the length of the opposite
131. 5:14side, without changing the denominator,
132. 5:17which is the length of the hypotenuse?
133. 5:20Well, [music] we would have to change
134. 5:21the angle alpha. Meaning, if we set the
135. 5:25hypotenuse to [music] a fixed length and
136. 5:27increase or decrease the angle alpha,
137. 5:30then the ratios [music] change. So, if
138. 5:33these ratios only change when alpha
139. 5:35changes, [music] then we can also
140. 5:37understand them as functions that depend
141. 5:39on alpha.
142. 5:41Each of these [music] functions has been
143. 5:42given its own name. The first one is
144. 5:45called s the next one cosine. Then
145. 5:48[music] there's tangent, co-angent,
146. 5:51seccant and cosecant.
147. 5:54If we [music] choose a fixed length for
148. 5:55the hypotenuse, then we might as well
149. 5:58choose a length that simplifies
150. 6:00everything a little bit. For instance,
151. 6:02with a length of one, the sign of alpha
152. 6:04simply corresponds to the length of the
153. 6:06opposite side and the cosine to the
154. 6:08length of the adjacent side. And even
155. 6:11better we [music] see that we can
156. 6:13calculate the other ratios using just
157. 6:15the sign and cosine. [music] In other
158. 6:18words, if we understand these two, we
159. 6:20also understand the others. And [music]
160. 6:23we can now also use our unit circle
161. 6:25template to measure the angle alpha.
162. 6:35All right. Now we need to visualize
163. 6:36these functions. [music]
164. 6:38Let's start with the sign. To do this,
165. 6:40let's draw a coordinate system and plot
166. 6:43the angle on the x-axis and the length
167. 6:45of the opposite side on the y-axis.
168. 6:47[music]
169. 6:49If we increase the angle slightly, the
170. 6:52opposite side becomes longer. As we
171. 6:54approach an angle of zero, the [music]
172. 6:56length becomes smaller and smaller. At
173. 6:59an angle of exactly zero, the opposite
174. 7:01side disappears. But we can imagine that
175. 7:04it's still there just with a length of
176. 7:06zero. If we now rotate around the entire
177. 7:10circle and trace the path, we obtain
178. 7:12this curve for the sign. Here, negative
179. 7:15lengths mean [music] that the opposite
180. 7:17side is in the lower semicircle.
181. 7:20And that looks like a wave, doesn't it?
182. 7:23For the cosine, [music] we can simply
183. 7:25copy the coordinate system and offset it
184. 7:27by 90° because the adjacent side is
185. 7:30offset by 90° [music]
186. 7:31to the opposite side.
187. 7:41However, we can also draw the cosine
188. 7:43function in the same coordinate [music]
189. 7:44system. Here we can also see that the
190. 7:47cosine looks exactly like the sign.
191. 7:48[music]
192. 7:49It's just shifted by 90° or half of pi.
193. 7:53So if we understand the sign, [music] we
194. 7:56also understand the cosine. And finally,
195. 8:00we can extend the sign function to a
196. 8:02larger angle range.
197. 8:03>> [music]
198. 8:03>> We can just agree that we can keep
199. 8:05rotating around the circle beyond 2 pi.
200. 8:09And a negative angle just means we
201. 8:11rotate clockwise. [music]
202. 8:17[music]
203. 8:20And with that we have managed to develop
204. 8:22a mathematical description for a wave
205. 8:25out of a triangle. In reality, of
206. 8:28course, not all waves look like this
207. 8:30pure [music] sine wave. But we can
208. 8:32manipulate this function with additional
209. 8:34numbers. For example, if we multiply the
210. 8:37sign by a certain number, we influence
211. 8:39the height of a wave crest, the
212. 8:41so-called amplitude. And if we multiply
213. 8:44the angle by a certain number, we
214. 8:46influence its frequency. [music] We can
215. 8:49also shift the function in the x or y
216. 8:52direction or even put alpha into a
217. 8:54separate function.
218. 8:58>> [music]
219. 9:03[music]