Trigonometric Functions - Various Artists

[00:00.0]Trigonometric Functions - Various Artists
[00:03.37]
[00:03.37]原唱：映射者/天儿
[00:05.13]
[00:05.13]后期：昔染
[00:06.22]
[00:06.22]视频：讲不清
[00:10.53]
[00:10.53]When you first study math about 1234
[00:12.96]你初次学习数学是从1 2 3 4开始的
[00:12.96]First study equation about xyzt
[00:14.95]你初次学习方程是从包含x y z t的方程式开始的
[00:14.95]It will help you to think in a logical way
[00:17.02]这有助你进行逻辑思考
[00:17.02]When you sing sine cosine tangent
[00:19.28]当你唱着 正弦 余弦 余弦 正切
[00:19.28]Sine cosine tangent cotangent
[00:21.47]正弦 余弦 正切 余切
[00:21.47]Sine cosine secant cosecant
[00:23.59]正弦 余弦 正割 余割
[00:23.59]Let's sing a song about trig-functions
[00:25.73]让我们唱起这首三角函数之歌
[00:25.73]sin(2π+α)=sinα
[00:27.68]
[00:27.68]cos(2π+α)=cosα
[00:30.02]
[00:30.02]tan(2π+α)=tanα
[00:31.96]
[00:31.96]Which is induction formula1 and induction formula 2
[00:34.35]这是第一类诱导公式 接下来是第二类诱导公式
[00:34.35]sin(π+α)=-sinα
[00:36.34]
[00:36.34]cos(π+α)=-cosα
[00:38.62]
[00:38.62]tan(π+α)=tanα
[00:40.68]
[00:40.68]sin(π-α)=sinα
[00:42.71]
[00:42.71]cos(π-α)=-cosα
[00:44.78]
[00:44.78]tan(π-α)=-tanα
[00:47.37]
[00:47.37]These are all those "name do not change"
[00:49.3]这些都是函数名不变
[00:49.3]As pi goes to half pi the difference shall be huge
[00:51.58]当π值缩小一半 结果会大不相同
[00:51.58]sin(π/2+α)=cosα
[00:53.35]
[00:53.35]sin(π/2-α)=cosα
[00:55.55]
[00:55.55]cos(π/2+α)=-sinα
[00:57.7]
[00:57.7]cos(π/2-α)=sinα
[00:59.89]
[00:59.89]tan(π/2+α)=-cotα
[01:02.02]
[01:02.02]tan(π/2-α)=cotα
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[01:08.65]That is to say the odds will change evens are conserved
[01:12.67]这就是说 奇变偶不变
[01:12.67]The notations that they get depend on where they are
[01:17.0]符号看象限
[01:17.0]But no matter where you are
[01:19.270004]可是无论你在哪里
[01:19.270004]I've gotta say that
[01:21.479996]我都会说
[01:21.479996]If you were my sine curve I'd be your cosine curve
[01:25.770004]如果你是正弦函数 我愿做你的余弦函数
[01:25.770004]I'll be your derivative you'll be my negative one
[01:30.01]我会成为你的导数 而你是我的负导数
[01:30.01]As you change you amplitude I change my phase
[01:34.08]当你改变振幅时 我会改变相位
[01:34.08]We can oscillate freely in the external space
[01:38.58]我们可以在外空间自由地波动
[01:38.58]As we change our period and costant at hand
[01:42.759995]当我们改变周期和身旁的常数时
[01:42.759995]We travel from the origin to infinity
[01:46.93]我们可以从原点一直到无穷尽
[01:46.93]It's you sine and you cosine
[01:51.41]正是你 正弦和余弦
[01:51.41]Who make charming music around the world
[01:55.479996]创造了这世上最动听的音乐
[01:55.479996]It's you tangent cotangent
[01:59.82]正是你 正切和余切
[01:59.82]Who proclaim the true meaning of centrosymmetry
[02:47.26]揭示了中心对称的真正含义
[02:47.26]You wanna measure width of a river height of a tower
[02:49.20999]你想要测量河流的宽度以及塔的高度
[02:49.20999]You scratch your head which cost you more than an hour
[02:51.41]你抓耳挠腮 冥思苦想了一小时也无济于事
[02:51.41]You don't need to ask any "gods" or" master" for help
[02:53.41]你无需向上帝或是伟人请求帮助
[02:53.41]This group of formulas are gonna help you solve
[02:55.76]以下这组公式让你的难题迎刃而解
[02:55.76]sin(α+β)=sinα•cosβ+cosα•sinβ
[02:58.99]
[02:58.99]cos(α+β)=cosα•cosβ-sinα•sinβ
[03:02.21]
[03:02.21]tan(α+β)=(tanα+tanβ)/(1-tanα•tanβ)
[03:06.44]
[03:06.44]sin(α-β)=sinα•cosβ-cosα•sinβ
[03:09.68]
[03:09.68]cos(α-β)=cosα•cosβ+sinα•sinβ
[03:12.85]
[03:12.85]tan(α-β)=(tanα-tanβ)/(1+tanα•tanβ)
[03:17.14]
[03:17.14]As you come across a right triangle you feel easy to solve
[03:20.17]当你遇到直角三角形时 你觉得很容易解决
[03:20.17]But an obtuse triange gonna make you feel confused
[03:22.59]可是钝角三角形让你感到困惑
[03:22.59]Don't worry about what you do
[03:23.65]不必担心 不必手足无措
[03:23.65]There are always means to solve
[03:24.53]总会找到解决办法
[03:24.53]As long as you master the sine cosine law
[03:30.12]只要你掌握了正余弦定理
[03:30.12]At this moment I've got nothing to say
[03:34.4]此刻我一言不发
[03:34.4]As trig-functions rain down upon me
[03:38.68]当三角函数如雨点一般落在我身上
[03:38.68]At this moment I've got nothing to say
[03:42.85]此刻我一言不发
[03:42.85]Let's sing a song about trig-functions
[03:47.22]让我们唱起这首三角函数之歌
[03:47.22]Long live the trigonometric functions
[03:52.022]三角函数万岁