三角函数的诱导公式三的推导过程，因基础差，求高手赐教！ 三角函数的诱导公式和推导过程
\u6c42\u4e09\u89d2\u51fd\u6570\u8bf1\u5bfc\u516c\u5f0f\u4e09\u7684\u63a8\u5bfc\u3002\u3002\u3002\u8c22\u8c22\u516c\u5f0f3 \u89d2\u03b1\u4e0e\u03c0-\u03b1\u7684\u6b63\u5f26\uff0c\u4f59\u5f26\u51fd\u6570\u5173\u7cfb \u89c2\u5bdf\u56fe\u5728\u5355\u4f4d\u5706\u4e2d\uff0c\u5f53\u2220MOP=\u03b1\u662f\u9510\u89d2\u65f6\uff0c\u4f5c\u2220MOP\u2018=\u03c0-\u03b1\uff0c\u4e0d\u96be\u770b\u51fa \u70b9P\u4e0eP'\u5173\u4e8ey\u8f74\u5bf9\u79f0\uff0c\u56e0\u6b64\uff0c\u5b83\u4eec\u7684\u7eb5\u5750\u6807\u76f8\u7b49\uff0c\u6a2a\u5750\u6807\u7684\u7edd\u5bf9\u503c\u76f8\u7b49\u4e14\u7b26\u53f7\u76f8\u53cd\uff08\u53ef\u4ee5\u9a8c\u8bc1\uff0c\u5f53\u03b1\u4e0d\u662f\u9510\u89d2\u65f6\uff0c\u8fd9\u4e00\u7ed3\u8bba\u4f9d\u7136\u6210\u7acb\uff09\u5373 \u5bf9\u4efb\u610f\u89d2\u03b1\u6709sin\uff08\u03c0-\u03b1\uff09=sin\u03b1\uff0ccos (\u03c0-\u03b1\uff09=-cos\u03b1
\u4e07\u80fd\u516c\u5f0f\u63a8\u5bfc
\u3000\u3000sin2\u03b1=2sin\u03b1cos\u03b1=2sin\u03b1cos\u03b1/(cos^2(\u03b1)+sin^2(\u03b1))......*\uff0c
\u3000\u3000\uff08\u56e0\u4e3acos^2(\u03b1)+sin^2(\u03b1)=1\uff09
\u3000\u3000\u518d\u628a*\u5206\u5f0f\u4e0a\u4e0b\u540c\u9664cos^2(\u03b1)\uff0c\u53ef\u5f97sin2\u03b1=2tan\u03b1/(1+tan^2(\u03b1))
\u3000\u3000\u7136\u540e\u7528\u03b1/2\u4ee3\u66ff\u03b1\u5373\u53ef\u3002
\u3000\u3000\u540c\u7406\u53ef\u63a8\u5bfc\u4f59\u5f26\u7684\u4e07\u80fd\u516c\u5f0f\u3002\u6b63\u5207\u7684\u4e07\u80fd\u516c\u5f0f\u53ef\u901a\u8fc7\u6b63\u5f26\u6bd4\u4f59\u5f26\u5f97\u5230\u3002
\u4e09\u500d\u89d2\u516c\u5f0f\u63a8\u5bfc
\u3000\u3000tan3\u03b1=sin3\u03b1/cos3\u03b1
\u3000\u3000=(sin2\u03b1cos\u03b1+cos2\u03b1sin\u03b1)/(cos2\u03b1cos\u03b1-sin2\u03b1sin\u03b1)
\u3000\u3000=(2sin\u03b1cos^2(\u03b1)+cos^2(\u03b1)sin\u03b1\uff0dsin^3(\u03b1))/(cos^3(\u03b1)\uff0dcos\u03b1sin^2(\u03b1)\uff0d2sin^2(\u03b1)cos\u03b1)
\u3000\u3000\u4e0a\u4e0b\u540c\u9664\u4ee5cos^3(\u03b1)\uff0c\u5f97\uff1a
\u3000\u3000tan3\u03b1=(3tan\u03b1\uff0dtan^3(\u03b1))/(1-3tan^2(\u03b1))
\u3000\u3000sin3\u03b1=sin(2\u03b1+\u03b1)=sin2\u03b1cos\u03b1+cos2\u03b1sin\u03b1
\u3000\u3000=2sin\u03b1cos^2(\u03b1)+(1\uff0d2sin^2(\u03b1))sin\u03b1
\u3000\u3000=2sin\u03b1\uff0d2sin^3(\u03b1)+sin\u03b1\uff0d2sin^3(\u03b1)
\u3000\u3000=3sin\u03b1\uff0d4sin^3(\u03b1)
\u3000\u3000cos3\u03b1=cos(2\u03b1+\u03b1)=cos2\u03b1cos\u03b1\uff0dsin2\u03b1sin\u03b1
\u3000\u3000=(2cos^2(\u03b1)\uff0d1)cos\u03b1\uff0d2cos\u03b1sin^2(\u03b1)
\u3000\u3000=2cos^3(\u03b1)\uff0dcos\u03b1+(2cos\u03b1\uff0d2cos^3(\u03b1))
\u3000\u3000=4cos^3(\u03b1)\uff0d3cos\u03b1
\u3000\u3000\u5373
\u3000\u3000sin3\u03b1=3sin\u03b1\uff0d4sin^3(\u03b1)
\u3000\u3000cos3\u03b1=4cos^3(\u03b1)\uff0d3cos\u03b1
\u548c\u5dee\u5316\u79ef\u516c\u5f0f\u63a8\u5bfc
\u3000\u3000\u9996\u5148,\u6211\u4eec\u77e5\u9053sin(a+b)=sina*cosb+cosa*sinb,sin(a-b)=sina*cosb-cosa*sinb
\u3000\u3000\u6211\u4eec\u628a\u4e24\u5f0f\u76f8\u52a0\u5c31\u5f97\u5230sin(a+b)+sin(a-b)=2sina*cosb
\u3000\u3000\u6240\u4ee5,sina*cosb=(sin(a+b)+sin(a-b))/2
\u3000\u3000\u540c\u7406,\u82e5\u628a\u4e24\u5f0f\u76f8\u51cf,\u5c31\u5f97\u5230cosa*sinb=(sin(a+b)-sin(a-b))/2
\u3000\u3000\u540c\u6837\u7684,\u6211\u4eec\u8fd8\u77e5\u9053cos(a+b)=cosa*cosb-sina*sinb,cos(a-b)=cosa*cosb+sina*sinb
\u3000\u3000\u6240\u4ee5,\u628a\u4e24\u5f0f\u76f8\u52a0,\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230cos(a+b)+cos(a-b)=2cosa*cosb
\u3000\u3000\u6240\u4ee5\u6211\u4eec\u5c31\u5f97\u5230,cosa*cosb=(cos(a+b)+cos(a-b))/2
\u3000\u3000\u540c\u7406,\u4e24\u5f0f\u76f8\u51cf\u6211\u4eec\u5c31\u5f97\u5230sina*sinb=-(cos(a+b)-cos(a-b))/2
\u3000\u3000\u8fd9\u6837,\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u79ef\u5316\u548c\u5dee\u7684\u56db\u4e2a\u516c\u5f0f:
\u3000\u3000sina*cosb=(sin(a+b)+sin(a-b))/2
\u3000\u3000cosa*sinb=(sin(a+b)-sin(a-b))/2
\u3000\u3000cosa*cosb=(cos(a+b)+cos(a-b))/2
\u3000\u3000sina*sinb=-(cos(a+b)-cos(a-b))/2
\u3000\u3000\u597d,\u6709\u4e86\u79ef\u5316\u548c\u5dee\u7684\u56db\u4e2a\u516c\u5f0f\u4ee5\u540e,\u6211\u4eec\u53ea\u9700\u4e00\u4e2a\u53d8\u5f62,\u5c31\u53ef\u4ee5\u5f97\u5230\u548c\u5dee\u5316\u79ef\u7684\u56db\u4e2a\u516c\u5f0f.
\u3000\u3000\u6211\u4eec\u628a\u4e0a\u8ff0\u56db\u4e2a\u516c\u5f0f\u4e2d\u7684a+b\u8bbe\u4e3ax,a-b\u8bbe\u4e3ay,\u90a3\u4e48a=(x+y)/2,b=(x-y)/2
\u3000\u3000\u628aa,b\u5206\u522b\u7528x,y\u8868\u793a\u5c31\u53ef\u4ee5\u5f97\u5230\u548c\u5dee\u5316\u79ef\u7684\u56db\u4e2a\u516c\u5f0f:
\u3000\u3000sinx+siny=2sin((x+y)/2)*cos((x-y)/2)
\u3000\u3000sinx-siny=2cos((x+y)/2)*sin((x-y)/2)
\u3000\u3000cosx+cosy=2cos((x+y)/2)*cos((x-y)/2)
\u3000\u3000cosx-cosy=-2sin((x+y)/2)*sin((x-y)/2)
sin(-a)=-sin(a)
cos(-a)=cos(a)
sin(π2-a)=cos(a)
cos(π2-a)=sin(a)
sin(π2+a)=cos(a)
cos(π2+a)=-sin(a)
sin(π-a)=sin(a)
cos(π-a)=-cos(a)
sin(π+a)=-sin(a)
cos(π+a)=-cos(a)
2.两角和与差的三角函数
sin(a+b)=sin(a)cos(b)+cos(α)sin(b)
cos(a+b)=cos(a)cos(b)-sin(a)sin(b)
sin(a-b)=sin(a)cos(b)-cos(a)sin(b)
cos(a-b)=cos(a)cos(b)+sin(a)sin(b)
tan(a+b)=tan(a)+tan(b)1-tan(a)tan(b)
tan(a-b)=tan(a)-tan(b)1+tan(a)tan(b)
3.和差化积公式
sin(a)+sin(b)=2sin(a+b2)cos(a-b2)
sin(a)−sin(b)=2cos(a+b2)sin(a-b2)
cos(a)+cos(b)=2cos(a+b2)cos(a-b2)
cos(a)-cos(b)=-2sin(a+b2)sin(a-b2)
4.二倍角公式
sin(2a)=2sin(a)cos(b)
cos(2a)=cos2(a)-sin2(a)=2cos2(a)-1=1-2sin2(a)
5.半角公式
sin2(a2)=1-cos(a)2
cos2(a2)=1+cos(a)2
tan(a2)=1-cos(a)sin(a)=sina1+cos(a)
6.万能公式
sin(a)=2tan(a2)1+tan2(a2)
cos(a)=1-tan2(a2)1+tan2(a2)
tan(a)=2tan(a2)1-tan2(a2)
7.其它公式(推导出来的 )
a⋅sin(a)+b⋅cos(a)=a2+b2sin(a+c) 其中 tan(c)=ba
a⋅sin(a)+b⋅cos(a)=a2+b2cos(a-c) 其中 tan(c)=ab
1+sin(a)=(sin(a2)+cos(a2))2
1-sin(a)=(sin(a2)-cos(a2))2手打的累死、
1、sin(-a)=-sina
sin(-a)=sin(0-a)=sin0cosa-sinacos0=0-sina=-sina
2、cos(-a)=cosa
cos(-a)=cos(0-a)=cos0cosa+sin0sina=cosa+0=cosa
3、sin(π/2-a)=cosa
sin(π/2-a)=sinπ/2cosa-sinacosπ/2=cosa-0=cosa
绛旓細sin2(a2)=1-cos(a)2 cos2(a2)=1+cos(a)2 tan(a2)=1-cos(a)sin(a)=sina1+cos(a)6.涓囪兘鍏紡 sin(a)=2tan(a2)1+tan2(a2)cos(a)=1-tan2(a2)1+tan2(a2)tan(a)=2tan(a2)1-tan2(a2)7.鍏跺畠鍏紡(鎺ㄥ鍑烘潵鐨 )a⋅sin(a)+b⋅cos(a)=a2+b2sin(a+c) 鍏...
绛旓細鍏紡3 瑙伪涓幭-伪鐨勬寮︼紝浣欏鸡鍑芥暟鍏崇郴 瑙傚療鍥惧湪鍗曚綅鍦嗕腑锛屽綋鈭燤OP=伪鏄攼瑙掓椂锛屼綔鈭燤OP鈥=蟺-伪锛屼笉闅剧湅鍑 鐐筆涓嶱'鍏充簬y杞村绉帮紝鍥犳锛屽畠浠殑绾靛潗鏍囩浉绛夛紝妯潗鏍囩殑缁濆鍊肩浉绛変笖绗﹀彿鐩稿弽锛堝彲浠ラ獙璇侊紝褰撐变笉鏄攼瑙掓椂锛岃繖涓缁撹渚濈劧鎴愮珛锛夊嵆 瀵逛换鎰忚伪鏈塻in锛埾-伪锛=sin伪锛宑os (蟺-伪锛...
绛旓細鍥炵瓟濡傚浘锛
绛旓細涓夎鍑芥暟璇卞鍏紡鎺ㄥ杩囩▼ sin锛堬紞a锛夛紳锛峴ina锛宻in锛堬紞a锛夛紳sin锛0锛峚锛夛紳sin0cosa锛峴inacos0锛0锛峴ina锛濓紞sina锛沜os锛堬紞a锛夛紳cosa锛宑os锛堬紞a锛夛紳cos锛0锛峚锛夛紳cos0cosa锛媠in0sina锛漜osa锛0锛漜osa锛泂in锛埾锛2锛峚锛夛紳cosa锛宻in锛埾锛2锛峚锛夛紳sin蟺锛2cosa锛峴inacos蟺锛2锛漜osa锛0锛漜os...
绛旓細tan锛埾/2+伪锛=锛峜ot伪.cot锛埾/2+伪锛=锛峵an伪.杩樻湁涓嬪垪鍏紡锛歴in锛埾/2+伪锛=cos伪.cos锛埾/2+伪锛=鈥攕in伪.tan锛埾/2+伪锛=锛峜ot伪.cot锛埾/2+伪锛=锛峵an伪.sec锛埾/2+伪锛=锛峜sc伪.csc锛埾/2+伪锛=sec伪.
绛旓細鍏充簬璇卞鍏紡锛屾墍鏈夌殑鍏紡閮藉彲浠ュ綊绾充负锛氬鍙樺伓涓嶅彉锛岀鍙风湅璞￠檺銆傚鍙樺伓涓嶅彉锛氬嵆鐪嬒/2鍓嶇殑绯绘暟鏄鏁拌繕鏄伓鏁帮紝濡傛灉鏄伓鏁帮紝閭ｄ箞鍑芥暟鍚嶄笉鍙橈紝濡傛灉鏄鏁帮紝鍙樻垚瀹冪殑浣欏悕鍑芥暟锛宻in(3蟺/2+a)锛3鏄鏁版墍浠ュ彉涓篶os锛屽張濡俢ot(蟺+a),蟺=2*蟺/2锛2鏄伓鏁版墍浠ヤ笉鍙橈紝鍑芥暟鍚嶄粛涓篶ot銆
绛旓細鎺ㄧ畻鍏紡:3蟺/2卤伪涓幬辩殑涓夎鍑芥暟鍊间箣闂寸殑鍏崇郴:sin(3蟺/2+伪)=-cos伪 cos(3蟺/2+伪)=sin伪 tan(3蟺/2+伪)=-cot伪 cot(3蟺/2+伪)=-tan伪 sin(3蟺/2-伪)=-cos伪 cos(3蟺/2-伪)=-sin伪 tan(3蟺/2-伪)=cot伪 cot(3蟺/2-伪)=tan伪 璇卞鍏紡璁板繂鍙ｈ瘈:鈥滃鍙樺伓涓嶅彉,...
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绛旓細涓婇潰杩欎簺璇卞鍏紡鍙互姒傛嫭涓猴細瀵逛簬蟺/2*k 卤伪(k鈭圸)鐨涓夎鍑芥暟鍊硷紝鈶犲綋k鏄伓鏁版椂锛屽緱鍒拔辩殑鍚屽悕鍑芥暟鍊硷紝鍗冲嚱鏁板悕涓嶆敼鍙;鈶″綋k鏄鏁版椂锛屽緱鍒拔辩浉搴旂殑浣欏嚱鏁板硷紝鍗硈in鈫抍os;cos鈫抯in;tan鈫抍ot,cot鈫抰an.(濂囧彉鍋朵笉鍙)鐒跺悗鍦ㄥ墠闈㈠姞涓婃妸伪鐪嬫垚閿愯鏃跺師鍑芥暟鍊肩殑绗﹀彿銆(绗﹀彿鐪嬭薄闄)涓婅堪鐨勮蹇...
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