[00:00:00] trigonometric functions - 天儿
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[00:02:00] 词：天儿
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[00:03:00] 曲：天儿
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[00:10:00] When you first study math about 1234
[00:12:00] 你初次学习数学是从1 2 3 4开始的
[00:12:00] First study equation about xyzt
[00:14:00] 你初次学习方程是从包含x y z t的方程式开始的
[00:14:00] It will help you to think in a logical way
[00:16:00] 这有助你进行逻辑思考
[00:16:00] When you sing sine cosine cosine tangent
[00:19:00] 当你唱着 正弦 余弦 余弦 正切
[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] 让我们唱起这首三角函数之歌
[00:25:00] sin(2π+α)=sinα
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[00:27:00] cos(2π+α)=cosα
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[00:30:00] tan(2π+α)=tanα
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[00:31:00] Which is induction formula1 and induction formula 2
[00:34:00] 这是第一类诱导公式 接下来是第二类诱导公式
[00:34:00] sin(π+α)= -sinα
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[00:36:00] cos(π+α)=-cosα
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[00:38:00] tan(π+α)= tanα
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[00:40:00] sin(π-α)= sinα
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[00:42:00] cos(π-α)=-cosα
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[00:45:00] tan(π-α)=-tanα
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[00:47:00] These are all those \"name do not -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值缩小一半 结果会大不相同
[00:51:00] sin(π/2+α)=cosα
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[00:53:00] cos(π/2+α)=-sinα
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[00:55:00] tan(π/2+α)=-cotα
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[00:58:00] sin(π/2-α)=cosα
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[01:00:00] cos(π/2-α)=sinα
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[01:02:00] tan(π/2-α)=cotα
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[01:08:00] That is to say the odds will change evens are conserved
[01:12:00] 这就是说 奇变偶不变
[01:12:00] The notations that they get depend on where they are
[01:16:00] 符号看象限
[01:16:00] But no matter where you are
[01:19:00] 可是无论你在哪里
[01:19:00] I\'ve gotta say that
[01:21:00] 我都会说
[01:21:00] If you were my sine curve I\'d be your cosine curve
[01:25:00] 如果你是正弦函数 我愿做你的余弦函数
[01:25:00] I\'ll be your derivative you\'ll be my negative one
[01:30:00] 我会成为你的导数 而你是我的负导数
[01:30:00] As you change you amplitude I change my phase
[01:33:00] 当你改变振幅时 我会改变相位
[01:33:00] We can oscillate freely in the external space
[01:38:00] 我们可以在外空间自由地波动
[01:38:00] As we change our period and costant at hand
[01:42:00] 当我们改变周期和身旁的常数时
[01:42:00] We travel from the origin to infinity
[01:47:00] 我们可以从原点一直到无穷尽
[01:47:00] It\'s you sine and you cosine
[01:51:00] 正是你 正弦和余弦
[01:51:00] Who make charming music around the world
[01:55:00] 创造了这世上最动听的音乐
[01:55:00] It\'s you tangent cotangent
[01:59:00] 正是你 正切和余切
[01:59:00] Who proclaim the true meaning of centrosymmetry
[02:47:00] 揭示了中心对称的真正含义
[02:47:00] You wanna measure width of a river height of a tower
[02:49:00] 你想要测量河流的宽度以及塔的高度
[02:49:00] You scratch your head which cost you more than an hour
[02:51:00] 你抓耳挠腮 冥思苦想了一小时也无济于事
[02:51:00] You don\'t need to ask any \"gods\" or\" master\" for help
[02:53:00] 你无需向上帝或是伟人请求帮助
[02:53:00] This group of formulas are gonna help you solve
[02:55:00] 以下这组公式让你的难题迎刃而解
[02:55:00] sin(α+β)=sinα•cosβ+cosα•sinβ
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[02:59:00] cos(α+β)=cosα•cosβ-sinα•sinβ
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[03:02:00] tan(α+β)=(tanα+tanβ)/(1-tanα•tanβ)
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[03:06:00] sin(α-β)=sinα•cosβ-cosα•sinβ
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[03:09:00] cos(α-β)=cosα•cosβ+sinα•sinβ
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[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 solve
[03:20:00] 当你遇到直角三角形时 你觉得很容易解决
[03:20:00] But an obtuse triangle gonna make you feel confused
[03:22:00] 可是钝角三角形让你感到困惑
[03:22:00] Don\'t worry about what you do
[03:23:00] 不必担心 不必手足无措
[03:23:00] There are always means to solve
[03:24:00] 总会找到解决办法
[03:24:00] As long as you master the sine cosine law
[03:30:00] 只要你掌握了正余弦定理
[03:30:00] At this moment I\'ve got nothing to say
[03:34:00] 此刻我一言不发
[03:34:00] As trig-functions rain down upon me
[03:38:00] 当三角函数如雨点一般落在我身上
[03:38:00] At this moment I\'ve got nothing to say
[03:42:00] 此刻我一言不发
[03:42:00] Let\'s sing a song about trig-functions
[03:47:00] 让我们唱起这首三角函数之歌
[03:47:00] Long live the trigonometric functions
[03:52:00] 三角函数万岁