[00:00:00] 三角函数之歌 - 天儿
[00:10:00] TME享有本翻译作品的著作权
[00:10:00] When you first study math about 1234
[00:12:00] 当你初学数学时 要先从最简单的数字开始
[00:12:00] First study equation about xyzt
[00:14:00] 然后深入了解方程式 搞懂xyzt的概念
[00:14:00] It will help you to think in a logical
[00:16:00] 这将帮助你提升逻辑思维
[00:16:00] Way
[00:16:00] 能力
[00:16:00] When you sing sine cosine cosine tangent
[00:19:00] 当你学会了正弦 余弦 正切和余切
[00:19:00] Sine cosine tangent cotangent
[00:21:00] 正弦 余弦 正切 余切
[00:21:00] Sine cosine secant cosecant
[00:23:00] 正弦 余弦 正割 余割
[00:23:00] Let\'s sing a song about trig functions
[00:25:00] 让我们来用一首歌记住这些三角函数吧
[00:25:00] sin（2π+α）=sinα
[00:27:00]    
[00:27:00] cos（2π+α）=cosα
[00:30:00]    
[00:30:00] tan（2π+α）=tanα
[00:31:00]    
[00:31:00] Which is induction formula1
[00:33:00] 这些就是三角函数的
[00:33:00] And induction formula 2
[00:34:00] 周期性规律
[00:34:00] sin（π+α）= —sinα
[00:36:00]    
[00:36:00] cos（π+α）=—cosα
[00:38:00]    
[00:38:00] tan（π+α）= tanα
[00:40:00]    
[00:40:00] sin（π－α）= sinα
[00:42:00]    
[00:42:00] cos（π－α）=－cosα
[00:45:00]    
[00:45:00] tan（π－α）=－tanα
[00:47:00]    
[00:47:00] These are all those name donot change
[00:49:00] 这些都是函数名不变
[00:49:00] As pi goes to half pi the difference shall be huge
[00:51:00] 当π成为π/2时 结果就会有显著的变化
[00:51:00] sin（π/2+α）=cosα
[00:53:00]    
[00:53:00] sin（π/2－α）=cosα
[00:55:00]    
[00:55:00] cos（π/2+α）=－sinα
[00:57:00]    
[00:57:00] cos（π/2－α）=sinα
[01:00:00]    
[01:00:00] tan（π/2+α）=－cotα
[01:02:00]    
[01:02:00] tan（π/2－α）=cotα
[01:08:00]    
[01:08:00] That is to say the odds will change evens are conserved
[01:12:00] 也就是说 奇变偶不变
[01:12:00] The notations that they get depend on where they are
[01:17:00] 符号看象限
[01:17:00] But no matter where you are
[01:19:00] 但不管你在哪
[01:19:00] I‘ve gotta say that
[01:21:00] 我想对你说
[01:21:00] If you were sine curve I\'d be your cosine curve
[01:25:00] 若你愿做我的正弦曲线 那我就做你的余弦曲线
[01:25:00] I\'ll be your derivative you\'ll be my negtive one
[01:29:00] 我是你的导数 你就是我的负导数
[01:29:00] As you change you amplitude I change my phase
[01:34:00] 当你改变振幅时 我也会改变相位
[01:34:00] We can oscillate freely in the external space
[01:38:00] 我们可以在外界空间自由振荡
[01:38:00] As we change our period and costant at hand
[01:42:00] 当我们改变周期和常数时
[01:42:00] We travel from the origin to infinity
[01:46:00] 我们可以从原点走到无穷远
[01:46:00] It\'s you sine and you cosine
[01:51:00] 你就是那正弦 余弦
[01:51:00] Who make charming music around the world
[01:55:00] 是你创造了动人的音乐
[01:55:00] It\'s you tangent cotangent
[01:59:00] 你就是那正切 余切
[01:59:00] Who proclaim the true meaning of centrosymmetry
[02:47:00] 你揭晓了中心对称的真谛
[02:47:00] You wanna measure width of a river height of a tower
[02:49:00] 你想测量河流的宽度或塔楼的高度时
[02:49:00] You scratch your head which cost you more than an hour
[02:51:00] 你抓耳挠腮 你冥思苦想了一个多小时
[02:51:00] You don\'t need to ask any \"gods\" or\" master\" for help
[02:53:00] 但你无需向大神们请教
[02:53:00] This group of formulas are gonna help you solve
[02:55:00] 因为下面这组公式就能帮你解决
[02:55:00] sin(α+β)=sinα•cosβ+cosα•sinβ
[02:58:00]    
[02:58:00] cos(α+β)=cosα•cosβ-sinα•sinβ
[03:02:00]    
[03:02:00] tan(α+β)=(tanα+tanβ)/(1-tanα•tanβ)
[03:06:00]    
[03:06:00] sin(α-β)=sinα•cosβ-cosα•sinβ
[03:08:00]    
[03:08:00] cos(α-β)=cosα•cosβ+sinα•sinβ
[03:12:00]    
[03:12:00] tan(α-β)=(tanα-tanβ)/(1+tanα•tanβ)
[03:18:00]    
[03:18:00] As you come across a right triangle you fell easy to sovle
[03:20:00] 当你遇到直角三角形时 你很容易就能解决
[03:20:00] But an obtuse triange\'s gonna make you feel confused
[03:22:00] 但钝角三角形可能会让你感到困惑
[03:22:00] Don\'t worry about what you do
[03:23:00] 你也无需焦虑
[03:23:00] There are always means to solve
[03:24:00] 只要你掌握了正弦余弦定律
[03:24:00] As long as you master the sine cosine law
[03:30:00] 总会有方法解决的
[03:30:00] At this momnet i\'ve got nothing to say
[03:34:00] 此刻 我已无需多说
[03:34:00] As trig functions rain down upon me
[03:38:00] 因为三角函数的内容真的太多了
[03:38:00] At this momnet i\'ve got nothing to say
[03:42:00] 此刻 我已无需多说
[03:42:00] Let\'s sing a song about trig functios
[03:47:00] 让我们来用一首歌记住这些三角函数吧
[03:47:00] Long live the trigonometric functions
[03:52:00] 三角函数万岁