三角函数的和差化乘公式是什么? 三角函数的和差化积公式
\u4e09\u89d2\u51fd\u6570\u90a3\u4e00\u5757\uff0c\u548c\u5dee\u5316\u79ef\u4e0e\u79ef\u5316\u548c\u5dee\u7684\u516c\u5f0f\u662f\u4ec0\u4e48\uff1f\u4e09\u89d2\u51fd\u6570\uff0c\u79ef\u5316\u548c\u5dee\u4e0e\u548c\u5dee\u5316\u79ef\uff0c\u63a8\u5bfc\u8fc7\u7a0b
sinx+siny=2sin((x+y)/2)*cos((x-y)/2)
sinx-siny=2cos((x+y)/2)*sin((x-y)/2)
cosx+cosy=2cos((x+y)/2)*cos((x-y)/2)
cosx-cosy=-2sin((x+y)/2)*sin((x-y)/2)
\u5bf9\u4e8e\u548c\u5dee\u5316\u79ef\u516c\u5f0f\u6765\u8bf4\uff0c\u82e5\u7b49\u53f7\u5de6\u8fb9\u5168\u662fsin\uff0c\u5219\u53f3\u8fb9\u5f02\u540d\uff0c\u82e5\u7b49\u53f7\u5de6\u8fb9\u5168\u662fcos\uff0c\u5219\u7b49\u53f7\u53f3\u8fb9\u540c\u540d\uff0c\u82e5\u7b49\u53f7\u5de6\u8fb9\u4e2d\u95f4\u7684\u6b63\u8d1f\u53f7\u51b3\u5b9a\u4e86\u53f3\u8fb9\u7b2c\u4e8c\u9879\uff0c\u82e5\u662f\u6b63\uff0c\u5219\u662fcos\uff0c\u82e5\u662f\u8d1f\uff0c\u5219\u662fsin\uff0c\u53ef\u4ee5\u6839\u636e\u7b2c\u4e00\u6761\u539f\u5219\u5199\u51fa\u5b8c\u6574\u7684\u53f3\u8fb9\u5f0f\u5b50\uff0c\u6700\u540e\u8bb0\u5f97cos-cos\u8981\u6dfb\u4e00\u4e2a\u8d1f\u53f7\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u4e09\u89d2\u51fd\u6570\u6982\u5ff5\u6ce8\u610f\u7684\u95ee\u9898\uff1a
1\u3001\u521d\u4e2d\u9636\u6bb5\u7684\u6240\u8bf4\u7684\u9510\u89d2\u4e09\u89d2\u51fd\u6570\u662f\u9510\u89d2\u7684\u6b63\u5f26\u3001\u4f59\u5f26\u3001\u6b63\u5207\u3001\u4f59\u5207\u56db\u79cd\u51fd\u6570\u7684\u7edf\u79f0\u3002
2\u3001\u9510\u89d2\u4e09\u89d2\u51fd\u6570\u8868\u793a\u7684\u662f\u4e24\u4e2a\u6b63\u6570\u7684\u6bd4\u503c\uff0c\u56e0\u800c\u9510\u89d2\u4e09\u89d2\u51fd\u6570\u6ca1\u6709\u5355\u4f4d\u3002
3\u3001\u7406\u6e05\u9510\u89d2\u4e09\u89d2\u51fd\u6570\u4e2d\u7684\u81ea\u53d8\u91cf\u4e0e\u56e0\u53d8\u91cf\uff0c\u5bf9\u4e8e\u56db\u79cd\u51fd\u6570\u6765\u8bf4\uff0c\u4ee5\u2220A\u4e3a\u4f8b\uff0c\u81ea\u53d8\u91cf\u90fd\u662f\u9510\u89d2A\uff0c\u56e0\u53d8\u91cf\u5c31\u662f\u9510\u89d2A\u7684\u56db\u79cd\u4e09\u89d2\u51fd\u6570\uff0c\u8fd9\u8bf4\u660e\u5f53\u9510\u89d2A\u7684\u5927\u5c0f\u4e0d\u53d8\u65f6\uff0c\u9510\u89d2A\u7684\u6b63\u5f26\u503c\u3001\u4f59\u5f26\u503c\u3001\u6b63\u5207\u503c\u3001\u4f59\u5207\u503c\u4e5f\u5c06\u4fdd\u6301\u4e0d\u53d8\u3002
4\u3001\u9510\u89d2\u4e09\u89d2\u51fd\u6570\u4e2d\u81ea\u53d8\u91cf\u7684\u53d6\u503c\u8303\u56f4\uff0c\u9510\u89d2\u4e09\u89d2\u51fd\u6570\u7684\u81ea\u53d8\u91cf\u662f\u9510\u89d2\uff0c\u6240\u4ee5\u81ea\u53d8\u91cf\u2220A\u7684\u8303\u56f4\u5c31\u662f0\u00b0\uff1c\u2220A\uff1c90\u00b0\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u4e09\u89d2\u51fd\u6570
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u548c\u5dee\u5316\u79ef
积化和差公式：
sinαsinβ=-[cos(α+β)-cos(α-β)]/2
cosαcosβ=[cos(α+β)+cos(α-β)]/2
sinαcosβ=[sin(α+β)+sin(α-β)]/2
cosαsinβ=[sin(α+β)-sin(α-β)]/2
和差化积公式：
sinθ+sinφ=2sin[(θ+φ)/2]cos[(θ-φ)/2]
sinθ-sinφ=2cos[(θ+φ)/2]sin[(θ-φ)/2]
cosθ+cosφ=2cos[(θ+φ)/2]cos[(θ-φ)/2]
cosθ-cosφ=-2sin[(θ+φ)/2]sin[(θ-φ)/2]
三角函数的和差化积公式
sinα＋sinβ＝2sin（α＋β）/2·cos（α－β）/2
sinα－sinβ＝2cos（α＋β）/2·sin（α－β）/2
cosα＋cosβ＝2cos（α＋β）/2·cos（α－β）/2
cosα－cosβ＝－2sin（α＋β）/2·sin（α－β）/2
三角函数的积化和差公式
sinα
·cosβ＝1/2
[sin（α＋β）＋sin（α－β）]
cosα
·sinβ＝1/2
[sin（α＋β）－sin（α－β）]
cosα
·cosβ＝1/2
[cos（α＋β）＋cos（α－β）]
sinα
·sinβ＝-1/2
[cos（α＋β）－cos（α－β）]
绛旓細鏄涓夎鍑芥暟鐨勫拰宸寲绉鍏紡鍚.绉寲鍜屽樊鍏紡锛歴in伪sin尾=-[cos(伪+尾)-cos(伪-尾)]/2 cos伪cos尾=[cos(伪+尾)+cos(伪-尾)]/2 sin伪cos尾=[sin(伪+尾)+sin(伪-尾)]/2 cos伪sin尾=[sin(伪+尾)-sin(伪-尾)]/2 鍜屽樊鍖栫Н鍏紡锛歴in胃+sin蠁=2sin[(胃+蠁)/2]cos[(胃-蠁...
绛旓細涓銆佷笁瑙掑嚱鏁颁箻绉彉鎹㈠拰宸叕寮 1銆乻inAsinB=-[cos(A+B)-cos(A-B)]/2銆2銆乧osAcosB=[cos(A+B)+cos(A-B)]/2銆3銆乻inAcosB=[sin(A+B)+sin(A-B)]/2銆4銆乧osAsinB=[sin(A+B)-sin(A-B)]/2銆備簩銆佷笁瑙掑嚱鏁板拰宸彉鎹箻绉叕寮 1銆乻inA+sinB=2sin[(A+B)/2]cos[(A-B)/2]銆...
绛旓細涓夎鍑芥暟鐨勫拰宸寲绉鍏紡鏄鐢ㄦ潵灏嗕袱涓笁瑙掑嚱鏁扮殑鍜屾垨宸浆鍖栦负涔樼Н鐨勫叕寮忋傚叿浣撳叕寮忓涓嬶細1. 涓夎鍑芥暟鐨勫拰宸寲绉叕寮忥細- 姝ｅ鸡鍑芥暟鐨勫拰宸寲绉叕寮忥細sin(A 卤 B) = sinA路cosB 卤 cosA路sinB - 浣欏鸡鍑芥暟鐨勫拰宸寲绉叕寮忥細cos(A 卤 B) = cosA路cosB ∓ sinA路sinB - 姝ｅ垏鍑芥暟鐨勫拰宸寲绉...
绛旓細涓夎鍑芥暟鐨勫拰宸寲绉叕寮忔槸鎸囧皢涓や釜涓夎鍑芥暟鐨勫拰鎴栧樊琛ㄧず涓轰竴涓笁瑙掑嚱鏁扮殑涔樼Н鐨勫叕寮銆1. 浣欏鸡鍑芥暟鐨勫拰宸寲绉叕寮忥細cos(A 卤 B) = cos(A) * cos(B) ∓ sin(A) * sin(B)2. 姝ｅ鸡鍑芥暟鐨勫拰宸寲绉叕寮忥細sin(A 卤 B) = sin(A) * cos(B) 卤 cos(A) * sin(B)杩欎簺鍏紡鍦ㄨВ...
绛旓細涓夎鍑芥暟鍜屽樊鍖栫Н鍏紡鏈塻in伪+sin尾=2sin[(伪+尾)/2]cos[(伪-尾)/2] sin伪-sin尾=2cos[(伪+尾)/2]sin[(伪-尾)/2] cos伪+cos尾=2cos[(伪+尾)/2]cos[(伪-尾)/2]绛夈備笁瑙掑嚱鏁扮殑鍜屽樊鍖栫Н鏄寚灏嗕袱涓笁瑙掑嚱鏁扮殑鍜屾垨宸浆鍖栦负涓涓笁瑙掑嚱鏁扮殑涔樼Н銆傝繖涓妧宸у湪瑙ｅ喅涓夎鍑芥暟鐨勮繍绠椼...
绛旓細涓夎鍑芥暟鐨勫拰宸寲绉鍏紡锛歴inA+sinB=2sin[(A+B)/2]cos[(A-B)/2]sinA-sinB=2cos[(A+B)/2]sin[(A-B)/2]cosA+cosB=2cos[(A+B)/2]cos[(A-B)/2]cosA-cosB=-2sin[(A+B)/2]sin[(A-B)/2]tanA+tanB=sin(A+B)/cosAcosB=tan(A+B)(1-tanAtanB)tanA-tanB=sin(A-B)/...
绛旓細sin(A - B) = sin A * cos B - cos A * sin B 3. 姝ｅ垏鍑芥暟鍜屽樊鍖绉鍏紡锛歵an(A + B) = (tan A + tan B) / (1 - tan A * tan B)tan(A - B) = (tan A - tan B) / (1 + tan A * tan B)杩欎簺鍏紡鍙互鐢ㄤ簬灏涓夎鍑芥暟鐨勫拰鎴栧樊杞崲鎴愪箻绉殑褰㈠紡锛屼粠鑰屾洿鏂逛究鍦...
绛旓細鍦ㄥ簲鐢鍜屽樊鍖绉鍏紡鏃讹紝鑻ョ瓑鍙峰乏杈圭殑鍑芥暟鍏ㄩ儴鏄痵in锛屽垯鍙宠竟鐨鍑芥暟鏄cos鍜宻in鐨勪箻绉傝嫢绛夊彿宸﹁竟鐨勫嚱鏁板叏閮ㄦ槸cos锛屽垯鍙宠竟鐨勫嚱鏁版槸cos鐨勪箻绉傝嫢绛夊彿宸﹁竟鍑芥暟涓棿鐨勬璐熷彿鍐冲畾浜嗗彸杈圭浜岄」鐨勫嚱鏁帮紝姝ｅ彿琛ㄧずcos锛岃礋鍙疯〃绀簊in銆傛牴鎹繖涓鍘熷垯锛屽彲浠ュ畬鏁村湴鍐欏嚭鍙宠竟鐨勫嚱鏁拌〃杈惧紡锛屽苟璁板緱鍦╟os鍜宑os鐩稿噺鏃舵坊鍔犱竴涓...
绛旓細涓夎鍑芥暟鍜岀Н鍖栧樊鍜屽樊鍖绉鍏紡濡備笅锛1銆佺Н鍖栧拰宸叕寮忔湁sin伪*cos尾=锛1/2锛塻in锛埼+尾锛+sin锛埼-尾锛夛紱cos伪*sin尾=锛1/2锛塻in锛埼+尾锛-sin锛埼-尾锛夛紱cos伪*cos尾=锛1/2锛塩os锛埼+尾锛+cos锛埼-尾锛夛紱sin伪*sin尾=锛1/2锛塩os锛埼+尾锛-cos锛埼-尾锛夈2銆佸樊鍖栫Н鍏紡鏈...
绛旓細cosx涔樹互cos3x =(cosx)^2*cos2x-sinx*sin2x*cosx =(cos2x+1)/2*cos2x-(sin2x)^2/2 =((cos2x)^2+1-(sin2x)^2)/2杩欎竴姝ュ悗闈㈡妸1鍙樹负 (sin2x)^2+(cos2x)^2 =(cos2x)^2 璁板繂鍙ｈ瘈 绉寲鍜屽樊寰楀拰宸紝浣欏鸡鍦ㄥ悗瑕佺浉鍔狅紱寮傚悕鍑芥暟鍙栨寮︼紝姝ｅ鸡鐩镐箻鍙栬礋鍙枫傝В閲婏細锛1锛夌Н鍖栧拰宸渶鍚...
