I would like to introduce to you a really general but very simple proof for all compound angle formulae with any positive angles a and b, which can be a > b or a < b. The starting formula should be cos (a - b). The unit circle is divided into 2 sectors by the two unit vectors having angles a and b. The length of the chord of the minor sector can be found by distance formula when the coordinates of the unit vectors are derived as x = cos a or b, and y = sin a or b. The reason for choosing cos (a - b) is that the angle between the unit radii forming one of the two sectors of unit circle is (a - b) when a > b, and - (a - b) when a < b, whatever the values of a and b. By rotational transformation about origin, the sector with this angle moves to a new position with original unit radius for angle a lying on the x-axis. The original unit radius for angle b will move to have a new angle = (a - b) or - (a - b). Hence a new set of coordinates for ends of the rotated chord can be established. A second distance equation for the chord length equal to that from the first distance equation can be established. With algebraic manipulation, the formula for cos (a - b) can be proved in the most general way. With this formula proven, other compound angle formulae can be derived by substituting (-b) for b, (90 + a) for a into the proven formula and substituting sin/cos for tan. I would like to call this method equal chords of rotated sectors method.
我媽問我為什麼跪著看手機
太感謝老師了
超清楚簡明，解惑，為何以前看到三角函數跟看到鬼一般
以前除了要處理數學，還要處理很多東西
I would like to introduce to you a really general but very simple proof for all compound angle formulae with any positive angles a and b, which can be a > b or a < b. The starting formula should be cos (a - b). The unit circle is divided into 2 sectors by the two unit vectors having angles a and b. The length of the chord of the minor sector can be found by distance formula when the coordinates of the unit vectors are derived as x = cos a or b, and y = sin a or b. The reason for choosing cos (a - b) is that the angle between the unit radii forming one of the two sectors of unit circle is (a - b) when a > b, and - (a - b) when a < b, whatever the values of a and b. By rotational transformation about origin, the sector with this angle moves to a new position with original unit radius for angle a lying on the x-axis. The original unit radius for angle b will move to have a new angle = (a - b) or - (a - b). Hence a new set of coordinates for ends of the rotated chord can be established. A second distance equation for the chord length equal to that from the first distance equation can be established. With algebraic manipulation, the formula for cos (a - b) can be proved in the most general way. With this formula proven, other compound angle formulae can be derived by substituting (-b) for b, (90 + a) for a into the proven formula and substituting sin/cos for tan. I would like to call this method equal chords of rotated sectors method.
很亲切的和差公式。上高一的数学老师已经给我们推导过了，那是30年前的事情了。很怀念那时年轻的学生生涯。
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感謝感謝感謝
學校老師都只會叫人死背
連多花幾分鐘證明一下都不願意
這真的對我很有幫助
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厲害 困惑十多年了 問GPT還給我瞎回答 謝謝老師🙏
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講得非常清楚非常簡潔!
太感謝了QQ 講的超好
謝謝 !! 好的觀眾才能成就好的影片~~
真是精彩呀！好像在看魔術
好強~這些方法倒底怎麼發現的
令為1的比例方法太棒了(但要用對地方 !？) ！非常簡明 ! 但我在國小(四十年多前)課本教的雞兔問題,也是令某某為1,但就是想不起是怎麼令的😅 ! 可否請老師,先進達人賜教 ! (現在在網上學到的方法是 , 四腳兔+兩腳雞除以2,變二腳兔+一腳雞,再減去總頭數,總頭數就是一腳兔+一腳雞的腳數,即變為一腳兔,而一腳兔的腳數就是兔數)
太感謝了! 我不記得高中時老師有推導給我們看(或許是自己沒注意聽)。 於是當時就靠背公式應付考題........ 怎麼可能學得好呢?
請問如何說明對於任意角alpha,beta，這些公式均成立？(即角alpha,beta不侷限於0至90度之間)
對，這似乎是幾何證明最大的侷限⋯⋯
可從幾何證明出發，設法逐步放寬角度限制、去除限制。但是在某些地方似乎很難完全越過，極傷腦筋。
還是把圖形架在直角座標平面上才好處理
very helpful in developing calculus from analytic geometry to quantum mechanics ❤
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講的很好欸，以前都是用背的
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請問能用類似的方法證明差角公式嗎?
講師說的數語譯聽不懂！
可以用簡單的白話文解釋嗎？謝謝
超級超級感謝
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6:59我一直搞不懂為什麼beta角的對邊等於tangent beta乘以1/cosine alpha
因為黃色三角形的底邊長度L是1/cosine α，所以(β角的)對邊就是tangent β/cosine α
因為 cos x tan = sin
謝謝老師~
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這個證明真是巧妙
謝謝肯定 !
身為國中生 我好後悔我以前怎不知道這部影片>_
謝謝支持 !
請問6:10為什麼是tan a 呢
謝謝你的提問~~因為按照定義，tan α = 對邊/鄰邊，而鄰邊的長度我們令其等於1，帶入前面的定義式，可以得到 tan α=對邊/1=對邊，所以對邊=tan α
Stepp學院 謝謝~目前即將升高一，正在努力預習物理中!
Cool~物理會用到很多數學，像是圓周運動和簡諧運動...等等，就是用三角函數去描述的。頻道以後也會陸續拍一些物理的影片，到時候歡迎多來逛~~
Stepp學院 好的好的~希望能在此頻道吸收到更多的物理知識~
睡不著就跑來看
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這證明太鬼了
So beautiful ! 可惜我高中時不知道這個
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😭😭😭……當年就是笨在令1除cosα得C斜長
至少我在考學測前還能看到這個，還不算太晚
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精妙的证明
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Proof without words
Yep, 所謂的"無字證明"
好強
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谢
🙏
讚
太複雜了啦 😖
great
高一生看的很好
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學那麼久 才知道怎麼來的
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學不會沒關係 出社會真的沒有用
除非你是當高中數學老師：）
不見得哦。🤔
@@a34-t2d 我目前是真的沒有用到的機會啦🤣
讚
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