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歡迎來學習弧度和角度這堂課

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我想你們應該對角度的概念

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都比較熟悉了

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在本課的關於角度的例子中

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我們做過了一些練習

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你們應該知道

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直角等於90度

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或者直角的一半等於45度

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你們也應該熟悉

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在圓中的一些概念

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這是我最擅長的畫圓的方法

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一個圓的角度是360度

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今天我將會給你們介紹

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另外一種衡量角的方法 或是說單位

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就是弧度

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弧度是什麽呢？

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我們先從定義開始吧

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我想這個會給你們一些直觀的認識

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爲什麽它被稱爲弧度

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我用一下畫圓的工具

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這樣畫的圓確實很不錯

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我還在用著弧度工具 不 是畫圓工具

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好的

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這個半徑的長度爲R

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弧度就是與一段圓弧相對的角

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“相對”意思是如果這是一個角

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這是一段圓弧

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這個角與這個圓弧相對

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這段圓弧與這個角相對

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所以一弧度

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1弧度等於與一段長度等於半徑的圓弧

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相對的角的大小

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所以這段弧的長度也等於R

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這個角就是1弧度

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這個塗的都不像樣了

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重新畫一個大一些的圓

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畫好了

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繼續來研究一下弧度是因爲我想知道

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爲什麽要使用弧度

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我們都知道角度

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不過考慮一下弧度

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就會覺得它是合理的

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用一下直線工具

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設定半徑長度爲R

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這段圓弧的長度也是R

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這個角 設爲θ

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等於1弧度

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現在就可以稱之爲1弧度了

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好像一個半徑一樣

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現在問一個問題：

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一個整圓一共有多少弧度？

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如果這段長是R

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那麽整個圓周的長度是多少？

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等於2πR 對嗎？

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根據基本的幾何知識就可以得出

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如果1弧度等於

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與長度爲R的圓弧相對的角的大小

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那麽與長爲2πR的圓弧相對的

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角的大小就是2π弧度

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所以這個角等於2π弧度

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如果你仍然沒明白 那就用這種方式考慮一下

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大小爲2π弧度的角

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與長爲2π乘以半徑的圓弧

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相對應

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我不知道半徑的複數形式怎麽念

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可能是radians吧

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我確實不知道

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所以爲什麽我通過這麽混亂的方式

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使你們更加困惑呢？

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我只是想給你們一個直觀的認識

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讓你們知道爲什麽稱之爲弧度

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同時也告訴你們弧度跟圓的關係

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現在知道一個圓周是2π弧度

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我們可以計算出

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弧度和角度之間的關係

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把這個刪了

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我們說過一個圓周爲2π弧度

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一個圓的角度是多少？

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繞圓一周是多少角度？

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當然是360度

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所以

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所以就可以建立一個

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弧度與角度之間的等式

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通過等式兩邊都除以2π

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所以1弧度等於360/(2π)度

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也就是180/π度

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同樣 我們可以用其他的方式來做

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可以在等式的兩邊都除以360

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我們就可以說1度

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兩邊都除以360

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我反過來寫

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1度等於2π除以360弧度

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也就是π比180弧度

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所以就有了轉換式：

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1弧度等於180比π度

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1度等於π比180弧度

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如果你容易經常忘記這個關係式

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其實記住這個也無妨

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如果確實容易忘記的話 可以回過頭來看這個

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2π弧度等於360度

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也可以用一種更簡單的

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方法

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可以考慮一下半圓

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一個半圓

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角度的大小是180度 對吧？

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這個是度數的符號

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我也可以把“度”寫出來

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它等於π弧度

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所以π弧度等於180度

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可以推導出這個數學式

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1弧度等於180/π度

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或1度等於π/180弧度

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我們來做幾個問題

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使你們可以對這個有一個直觀的認識

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如果問你45度

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將它轉化成弧度

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我們都知道1度等於π比180弧度

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所以45度等於

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45乘以π比180弧度

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我們來看一看 180除以45

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180除以45等於4

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所以等於π比4弧度

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45度等於π比4弧度

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把它記住

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這兩個僅僅是不同的單位

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或者是不同的度量角的方法

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我們之所以要這麽做就是爲了

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數學上測量角度的標準化

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盡管大部分人在日常生活中

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對於角度更熟悉一些

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我們來做一些其它的例子

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需要好好地記住：

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1弧度等於180比π度

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1度等於π比180弧度

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如果你們會混淆 就把這個寫下來

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我把它寫下來就是因爲

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我常常忘了到底是π比180還是180比π

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我僅僅記著π弧度等於180度就行啦

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讓我們來做另外一個例子

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如果我問π比2弧度

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等於多少度呢？

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我已經忘了我剛才寫的關係式了

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我僅僅記得

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π弧度等於180度

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我的妻子剛剛回家

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我需要現在

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離開課堂一會

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過會會繼續的

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還是讓我們先把問題解決了吧

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然後我再去見我妻子

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我們都知道π弧度等於180度

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對嗎？

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所以1弧度等於180比―

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1弧度等於180度比π度

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我剛剛把公式推算出來

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因爲我老是忘記公式

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我們再回到這裡

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所以π比2弧度等於

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π比2乘以180比π度

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最終等於90度

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我們再做一個例子

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30度

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我又忘記了公式 所以我只需要記住

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π弧度等於180度

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所以1度等於π比180弧度

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所以30度等於

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30乘以π比180弧度

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也就是 30比180爲6

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等於π/6弧度

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希望你們能領悟到

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如何將弧度和度進行轉換

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以及爲什麽稱它爲弧度

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因爲它跟半徑有關係

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那樣當別人問你時 你就應答自如了

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不知道是什麽問題 例如弧度和角度的轉換

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我們下一節課見