跪求三角函数诱导公式推导两角和与差的正弦余弦正切公式的推导过程带图的 诱导公式的推导过程
\u4e09\u89d2\u51fd\u6570\u7684\u8bf1\u5bfc\u516c\u5f0f\u548c\u63a8\u5bfc\u8fc7\u7a0b\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)
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\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 \u3000\u3000\u4e09\u500d\u89d2\u516c\u5f0f\u63a8\u5bfc \u3000\u3000tan3\u03b1\uff1dsin3\u03b1/cos3\u03b1 \u3000\u3000\uff1d(sin2\u03b1cos\u03b1\uff0bcos2\u03b1sin\u03b1)/(cos2\u03b1cos\u03b1-sin2\u03b1sin\u03b1) \u3000\u3000\uff1d(2sin\u03b1cos^2(\u03b1)\uff0bcos^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\uff1d(3tan\u03b1\uff0dtan^3(\u03b1))/(1-3tan^2(\u03b1)) \u3000\u3000sin3\u03b1\uff1dsin(2\u03b1\uff0b\u03b1)\uff1dsin2\u03b1cos\u03b1\uff0bcos2\u03b1sin\u03b1 \u3000\u3000\uff1d2sin\u03b1cos^2(\u03b1)\uff0b(1\uff0d2sin^2(\u03b1))sin\u03b1 \u3000\u3000\uff1d2sin\u03b1\uff0d2sin^3(\u03b1)\uff0bsin\u03b1\uff0d2sin^3(\u03b1) \u3000\u3000\uff1d3sin\u03b1\uff0d4sin^3(\u03b1) \u3000\u3000cos3\u03b1\uff1dcos(2\u03b1\uff0b\u03b1)\uff1dcos2\u03b1cos\u03b1\uff0dsin2\u03b1sin\u03b1 \u3000\u3000\uff1d(2cos^2(\u03b1)\uff0d1)cos\u03b1\uff0d2cos\u03b1sin^2(\u03b1) \u3000\u3000\uff1d2cos^3(\u03b1)\uff0dcos\u03b1\uff0b(2cos\u03b1\uff0d2cos^3(\u03b1)) \u3000\u3000\uff1d4cos^3(\u03b1)\uff0d3cos\u03b1 \u3000\u3000\u5373 \u3000\u3000sin3\u03b1\uff1d3sin\u03b1\uff0d4sin^3(\u03b1) \u3000\u3000cos3\u03b1\uff1d4cos^3(\u03b1)\uff0d3cos\u03b1 \u3000\u3000\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)
三角函数公式两角和公式
sin(A+B) = sinAcosB+cosAsinB
sin(A-B) = sinAcosB-cosAsinB
cos(A+B) = cosAcosB-sinAsinB
cos(A-B) = cosAcosB+sinAsinB
tan(A+B) = (tanA+tanB)/(1-tanAtanB)
tan(A-B) = (tanA-tanB)/(1+tanAtanB)
cot(A+B) = (cotAcotB-1)/(cotB+cotA)
cot(A-B) = (cotAcotB+1)/(cotB-cotA)
倍角公式
tan2A = 2tanA/(1-tan^2 A)
Sin2A=2SinA?CosA
Cos2A = Cos^2 A--Sin^2 A
=2Cos^2 A—1
=1—2sin^2 A
三倍角公式
sin3A = 3sinA-4(sinA)^3;
cos3A = 4(cosA)^3 -3cosA
tan3a = tan a ? tan(π/3+a)? tan(π/3-a)
半角公式
sin(A/2) = √{(1--cosA)/2}
cos(A/2) = √{(1+cosA)/2}
tan(A/2) = √{(1--cosA)/(1+cosA)}
cot(A/2) = √{(1+cosA)/(1-cosA)}
tan(A/2) = (1--cosA)/sinA=sinA/(1+cosA)
和差化积
sin(a)+sin(b) = 2sin[(a+b)/2]cos[(a-b)/2]
sin(a)-sin(b) = 2cos[(a+b)/2]sin[(a-b)/2]
cos(a)+cos(b) = 2cos[(a+b)/2]cos[(a-b)/2]
cos(a)-cos(b) = -2sin[(a+b)/2]sin[(a-b)/2]
tanA+tanB=sin(A+B)/cosAcosB
积化和差
sin(a)sin(b) = -1/2*[cos(a+b)-cos(a-b)]
cos(a)cos(b) = 1/2*[cos(a+b)+cos(a-b)]
sin(a)cos(b) = 1/2*[sin(a+b)+sin(a-b)]
cos(a)sin(b) = 1/2*[sin(a+b)-sin(a-b)]
诱导公式
sin(-a) = -sin(a)
cos(-a) = cos(a)
sin(π/2-a) = cos(a)
cos(π/2-a) = sin(a)
sin(π/2+a) = cos(a)
cos(π/2+a) = -sin(a)
sin(π-a) = sin(a)
cos(π-a) = -cos(a)
sin(π+a) = -sin(a)
cos(π+a) = -cos(a)
tgA=tanA = sinA/cosA
万能公式
sin(a) = [2tan(a/2)] / {1+[tan(a/2)]^2}
cos(a) = {1-[tan(a/2)]^2} / {1+[tan(a/2)]^2}
tan(a) = [2tan(a/2)]/{1-[tan(a/2)]^2}
其它公式
a?sin(a)+b?cos(a) = [√(a^2+b^2)]*sin(a+c) [其中，tan(c)=b/a]
a?sin(a)-b?cos(a) = [√(a^2+b^2)]*cos(a-c) [其中，tan(c)=a/b]
1+sin(a) = [sin(a/2)+cos(a/2)]^2;
1-sin(a) = [sin(a/2)-cos(a/2)]^2;;
其他非重点三角函数
csc(a) = 1/sin(a)
sec(a) = 1/cos(a)
双曲函数
sinh(a) = [e^a-e^(-a)]/2
cosh(a) = [e^a+e^(-a)]/2
tg h(a) = sin h(a)/cos h(a)
公式一：
设α为任意角，终边相同的角的同一三角函数的值相等：
sin（2kπ＋α）= sinα
cos（2kπ＋α）= cosα
tan（2kπ＋α）= tanα
cot（2kπ＋α）= cotα
公式二：
设α为任意角，π+α的三角函数值与α的三角函数值之间的关系：
sin（π＋α）= -sinα
cos（π＋α）= -cosα
tan（π＋α）= tanα
cot（π＋α）= cotα
公式三：
任意角α与 -α的三角函数值之间的关系：
sin（-α）= -sinα
cos（-α）= cosα
tan（-α）= -tanα
cot（-α）= -cotα
公式四：
利用公式二和公式三可以得到π-α与α的三角函数值之间的关系：
sin（π-α）= sinα
cos（π-α）= -cosα
tan（π-α）= -tanα
cot（π-α）= -cotα
公式五：
利用公式-和公式三可以得到2π-α与α的三角函数值之间的关系：
sin（2π-α）= -sinα
cos（2π-α）= cosα
tan（2π-α）= -tanα
cot（2π-α）= -cotα
公式六：
π/2±α及3π/2±α与α的三角函数值之间的关系：
sin（π/2+α）= cosα
cos（π/2+α）= -sinα
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绛旓細鍘熺悊锛涓夎鍑芥暟鍊间腑锛屾寮︿竴浜岃薄闄愪负姝ｏ紝浣欏鸡涓鍥涜薄闄愪负姝ｏ紝姝ｅ垏涓涓夎薄闄愪负姝ｏ紙缁堣竟锛(5)sin(蟺/2+x)=cosx cos(蟺/2+x)=-sinx tan(蟺/2+x)=-cotx (6)sin(蟺/2-x)=cosx cos(蟺/2-x)=sinx tan(蟺/2-x)=cotx 涓よ鍏紡 (1)涓よ鍜宸叕寮 sin(x+y)=sinxcosy+sinycosx sin(x-y...
绛旓細涓よ鍜涓庡樊鐨涓夎鍑芥暟鍏紡 涓囪兘鍏紡 sin锛埼憋紜尾锛夛紳sin伪cos尾锛媍os伪sin尾 sin锛埼憋紞尾锛夛紳sin伪cos尾锛峜os伪sin尾 cos锛埼憋紜尾锛夛紳cos伪cos尾锛峴in伪sin尾 cos锛埼憋紞尾锛夛紳cos伪cos尾锛媠in伪sin尾 tan伪锛媡an尾 tan锛埼憋紜尾锛夛紳鈥斺斺1锛峵an伪 路tan尾 tan伪锛峵an尾 tan锛埼憋紞尾锛...
绛旓細鍋囧鏈変竴涓洿瑙涓夎褰 ABC,鍏朵腑 a銆乥 鏄洿瑙掕竟锛宑 鏄枩杈广傛寮(sin)绛変簬瀵硅竟姣旀枩杈;sinA=a/c锛涗綑寮(cos)绛変簬閭昏竟姣旀枩杈;cosA=b/c锛涙鍒(tan)绛変簬瀵硅竟姣旈偦杈;tanA=a/b銆
绛旓細鍏ㄩ兘鍙互鐢涓よ鍜涓庡樊鐨涓夎鍑芥暟鍏紡灞曞紑銆 鐑績缃戝弸| 鍙戝竷浜2011-06-09 涓炬姤| 璇勮 0 0 鍏堣鍏紡涓: 璁疚变负浠绘剰瑙,缁堣竟鐩稿悓鐨勮鐨勫悓涓涓夎鍑芥暟鐨勫肩浉绛: sin(2k蟺+伪)=sin伪 k鈭坺 cos(2k蟺+伪)=cos伪 k鈭坺 tan(2k蟺+伪)=tan伪 k鈭坺 cot(2k蟺+伪)=cot伪 k鈭坺鎺ㄥ杩囩▼鍏跺疄寰堢畝鍗,浣嗗湪...
绛旓細鍏跺畠鍏紡 a*sin(a)+b*cos(a)=sqrt(a^2+b^2)sin(a+c)[鍏朵腑锛宼an(c)=b/a]a*sin(a)-b*cos(a)=sqrt(a^2+b^2)cos(a-c)[鍏朵腑锛宼an(c)=a/b]1+sin(a)=(sin(a/2)+cos(a/2))^2 1-sin(a)=(sin(a/2)-cos(a/2))^2 鍏朵粬闈為噸鐐涓夎鍑芥暟 csc(a)=1/sin(a)sec(a...
绛旓細甯哥敤鐨璇卞鍏紡鏈変互涓嬪嚑缁勶細1.sin伪^2 +cos伪^2=12.sin伪/cos伪=tan伪3.tan伪=1/cot伪鍏紡涓:璁疚变负浠绘剰瑙掞紝缁堣竟鐩稿悓鐨勮鐨勫悓涓涓夎鍑芥暟鐨勫肩浉绛夛細sin锛2k蟺+伪锛=sin伪cos锛2k蟺+伪锛=cos伪tan锛2k蟺+伪锛=tan伪cot锛2k蟺+伪锛=cot伪鍏紡浜岋細璁疚变负浠绘剰瑙掞紝蟺+伪鐨勪笁瑙掑嚱鏁板间笌伪...
绛旓細璇卞鍏紡5鍜6濡備笅锛涓夎鍑芥暟璇卞鍏紡鏄暟瀛﹀叕寮忥紝鎸囦笁瑙掑嚱鏁颁腑锛屽埄鐢ㄥ懆鏈熸у皢瑙掑害姣旇緝澶х殑涓夎鍑芥暟锛岃浆鎹负瑙掑害姣旇緝灏忕殑涓夎鍑芥暟鐨勫叕寮忥紝鍏紡鏈夊叚缁勶紝鍏54涓備笁瑙掑嚱鏁拌瀵煎叕寮忔槸灏嗚n路(蟺/2)卤伪鐨勪笁瑙掑嚱鏁拌浆鍖栦负瑙捨辩殑涓夎鍑芥暟锛屽寘鎷竴浜涘父鐢ㄧ殑鍏紡鍜屽拰宸寲绉叕寮忋傚叕寮忎簲锛氬埄鐢ㄥ叕寮忎竴鍜屽叕寮忎笁鍙互...
绛旓細鎺ㄧ畻鍏紡锛3蟺/2卤伪涓幬辩殑涓夎鍑芥暟鍊间箣闂寸殑鍏崇郴锛歴in锛3蟺/2+伪锛=锛峜os伪 sin锛3蟺/2锛嵨憋級=锛峜os伪 cos锛3蟺/2+伪锛=sin伪 cos锛3蟺/2锛嵨憋級=锛峴in伪 tan锛3蟺/2+伪锛=锛峜ot伪 璇卞鍏紡璁板繂鍙ｈ瘈锛氣滃鍙樺伓涓嶅彉锛岀鍙风湅璞￠檺鈥濄傗滃銆佸伓鈥濇寚鐨勬槸蟺/2鐨勫嶆暟鐨勫鍋讹紝鈥滃彉涓庝笉...
