高中数学三角函数 高中数学三角函数(完整加分)
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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)
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tan2A = 2tanA/(1-tan^2 A)
Sin2A=2SinA?CosA
Cos2A = Cos^2 A--Sin^2 A
=2Cos^2 A\u20141
=1\u20142sin^2 A
\u4e09\u500d\u89d2\u516c\u5f0f
sin3A = 3sinA-4(sinA)^3;
cos3A = 4(cosA)^3 -3cosA
tan3a = tan a ? tan(\u03c0/3+a)? tan(\u03c0/3-a)
\u534a\u89d2\u516c\u5f0f
sin(A/2) = \u221a{(1--cosA)/2}
cos(A/2) = \u221a{(1+cosA)/2}
tan(A/2) = \u221a{(1--cosA)/(1+cosA)}
cot(A/2) = \u221a{(1+cosA)/(1-cosA)}
tan(A/2) = (1--cosA)/sinA=sinA/(1+cosA)
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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
\u79ef\u5316\u548c\u5dee
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)]
\u8bf1\u5bfc\u516c\u5f0f
sin(-a) = -sin(a)
cos(-a) = cos(a)
sin(\u03c0/2-a) = cos(a)
cos(\u03c0/2-a) = sin(a)
sin(\u03c0/2+a) = cos(a)
cos(\u03c0/2+a) = -sin(a)
sin(\u03c0-a) = sin(a)
cos(\u03c0-a) = -cos(a)
sin(\u03c0+a) = -sin(a)
cos(\u03c0+a) = -cos(a)
tgA=tanA = sinA/cosA
\u4e07\u80fd\u516c\u5f0f
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}
\u5176\u5b83\u516c\u5f0f
a?sin(a)+b?cos(a) = [\u221a(a^2+b^2)]*sin(a+c) [\u5176\u4e2d\uff0ctan(c)=b/a]
a?sin(a)-b?cos(a) = [\u221a(a^2+b^2)]*cos(a-c) [\u5176\u4e2d\uff0ctan(c)=a/b]
1+sin(a) = [sin(a/2)+cos(a/2)]^2;
1-sin(a) = [sin(a/2)-cos(a/2)]^2;;
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csc(a) = 1/sin(a)
sec(a) = 1/cos(a)
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sinh(a) = [e^a-e^(-a)]/2
cosh(a) = [e^a+e^(-a)]/2
tg h(a) = sin h(a)/cos h(a)
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sin\uff082k\u03c0\uff0b\u03b1\uff09= sin\u03b1
cos\uff082k\u03c0\uff0b\u03b1\uff09= cos\u03b1
tan\uff082k\u03c0\uff0b\u03b1\uff09= tan\u03b1
cot\uff082k\u03c0\uff0b\u03b1\uff09= cot\u03b1
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sin\uff08\u03c0\uff0b\u03b1\uff09= -sin\u03b1
cos\uff08\u03c0\uff0b\u03b1\uff09= -cos\u03b1
tan\uff08\u03c0\uff0b\u03b1\uff09= tan\u03b1
cot\uff08\u03c0\uff0b\u03b1\uff09= cot\u03b1
\u516c\u5f0f\u4e09\uff1a
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sin\uff08-\u03b1\uff09= -sin\u03b1
cos\uff08-\u03b1\uff09= cos\u03b1
tan\uff08-\u03b1\uff09= -tan\u03b1
cot\uff08-\u03b1\uff09= -cot\u03b1
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sin\uff08\u03c0-\u03b1\uff09= sin\u03b1
cos\uff08\u03c0-\u03b1\uff09= -cos\u03b1
tan\uff08\u03c0-\u03b1\uff09= -tan\u03b1
cot\uff08\u03c0-\u03b1\uff09= -cot\u03b1
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sin\uff082\u03c0-\u03b1\uff09= -sin\u03b1
cos\uff082\u03c0-\u03b1\uff09= cos\u03b1
tan\uff082\u03c0-\u03b1\uff09= -tan\u03b1
cot\uff082\u03c0-\u03b1\uff09= -cot\u03b1
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sin\uff08\u03c0/2+\u03b1\uff09= cos\u03b1
cos\uff08\u03c0/2+\u03b1\uff09= -sin\u03b1
∴A=2,T=2(7π/8-3π/8)=π
∴w=2π/π=2
∴f(x)=2sin(2x+q)
将x=3π/8,y=2代入:
2sin(3π/4+q)=2
sin(3π/4+q)=1
q属于(-π/2,π/2),
3π/4+q属于(π/4,5π/4)
3π/4+q=π/2
q=-π/4
f(x)=2sin(2x+π/4)
第二问:
f(A/2)=√2/5
2sin(A+π/4)=√2sinA+√2cosA=√2/5
sinA+cosA=1/5
sinA=1/5-cosA
sin²A=1/25-2/5cosA+cos²A
1-cos²A=1/25-2/5cosA+cos²A
cos²A-1/5cosA-12/25=0
(cosA+3/5)(cosA-4/5)=0
cosA=4/5>sinA+cosA,舍去
cosA=-3/5,sinA=4/5
∵4sinB=5sinC
又,根据正弦定理:b/sinB=c/sinC
∴sinB/sinC=5/4=b/c
∵a=6,令b=5t,c=4t
根据余弦定理a²=b²+c²-2bccosA有:
6²=25t²+16t²-2*5t*4t*(-3/5)=65t²
t=6√65/65
b=5t=6√65/13
c=4t=24√65/65
三角函数看似有很多,很复杂,但只要掌握了三角函数的本质及内部规律就能够发现三角函数各个公式之间有着强大的联系。而掌握三角函数的内部规律及本质也是学好三角函数的关键所在,下面是学习方法网为大家精心整理的三角函数公式大全:
锐角三角函数公式
sin α=∠α的对边 / 斜边
cos α=∠α的邻边 / 斜边
tan α=∠α的对边 / ∠α的邻边
cot α=∠α的邻边 / ∠α的对边
倍角公式
Sin2A=2SinA?CosA
Cos2A=CosA^2-SinA^2=1-2SinA^2=2CosA^2-1
tan2A=(2tanA)/(1-tanA^2)
(注:SinA^2 是sinA的平方 sin2(A) )
三倍角公式
sin3α=4sinα·sin(π/3+α)sin(π/3-α)
cos3α=4cosα·cos(π/3+α)cos(π/3-α)
tan3a = tan a · tan(π/3+a)· tan(π/3-a)
三倍角公式推导
sin3a
=sin(2a+a)
=sin2acosa+cos2asina
辅助角公式
Asinα+Bcosα=(A^2+B^2)^(1/2)sin(α+t),其中
sint=B/(A^2+B^2)^(1/2)
cost=A/(A^2+B^2)^(1/2)
tant=B/A
Asinα+Bcosα=(A^2+B^2)^(1/2)cos(α-t),(此括号内不是文章内容,来自学习方法网,阅读请跳过),tant=A/B
降幂公式
sin^2(α)=(1-cos(2α))/2=versin(2α)/2
cos^2(α)=(1+cos(2α))/2=covers(2α)/2
tan^2(α)=(1-cos(2α))/(1+cos(2α))
推导公式
tanα+cotα=2/sin2α
tanα-cotα=-2cot2α
1+cos2α=2cos^2α
1-cos2α=2sin^2α
1+sinα=(sinα/2+cosα/2)^2
=2sina(1-sin²a)+(1-2sin²a)sina
=3sina-4sin³a
cos3a
=cos(2a+a)
=cos2acosa-sin2asina
=(2cos²a-1)cosa-2(1-sin²a)cosa
=4cos³a-3cosa
sin3a=3sina-4sin³a
=4sina(3/4-sin²a)
=4sina[(√3/2)²-sin²a]
=4sina(sin²60°-sin²a)
=4sina(sin60°+sina)(sin60°-sina)
=4sina*2sin[(60+a)/2]cos[(60°-a)/2]*2sin[(60°-a)/2]cos[(60°-a)/2]
=4sinasin(60°+a)sin(60°-a)
cos3a=4cos³a-3cosa
=4cosa(cos²a-3/4)
=4cosa[cos²a-(√3/2)²]
=4cosa(cos²a-cos²30°)
=4cosa(cosa+cos30°)(cosa-cos30°)
=4cosa*2cos[(a+30°)/2]cos[(a-30°)/2]*{-2sin[(a+30°)/2]sin[(a-30°)/2]}
=-4cosasin(a+30°)sin(a-30°)
=-4cosasin[90°-(60°-a)]sin[-90°+(60°+a)]
=-4cosacos(60°-a)[-cos(60°+a)]
=4cosacos(60°-a)cos(60°+a)
上述两式相比可得
tan3a=tanatan(60°-a)tan(60°+a)
半角公式
tan(A/2)=(1-cosA)/sinA=sinA/(1+cosA);
cot(A/2)=sinA/(1-cosA)=(1+cosA)/sinA.
sin^2(a/2)=(1-cos(a))/2
cos^2(a/2)=(1+cos(a))/2
tan(a/2)=(1-cos(a))/sin(a)=sin(a)/(1+cos(a))
学习方法网[www.xuexidiyi.com]
三角和
sin(α+β+γ)=sinα·cosβ·cosγ+cosα·sinβ·cosγ+cosα·cosβ·sinγ-sinα·sinβ·sinγ
cos(α+β+γ)=cosα·cosβ·cosγ-cosα·sinβ·sinγ-sinα·cosβ·sinγ-sinα·sinβ·cosγ
tan(α+β+γ)=(tanα+tanβ+tanγ-tanα·tanβ·tanγ)/(1-tanα·tanβ-tanβ·tanγ-tanγ·tanα)
两角和差
cos(α+β)=cosα·cosβ-sinα·sinβ
cos(α-β)=cosα·cosβ+sinα·sinβ
sin(α±β)=sinα·cosβ±cosα·sinβ
tan(α+β)=(tanα+tanβ)/(1-tanα·tanβ)
tan(α-β)=(tanα-tanβ)/(1+tanα·tanβ)
和差化积
sinθ+sinφ = 2 sin[(θ+φ)/2] cos[(θ-φ)/2]
sinθ-sinφ = 2 cos[(θ+φ)/2] sin[(θ-φ)/2]
cosθ+cosφ = 2 cos[(θ+φ)/2] cos[(θ-φ)/2]
cosθ-cosφ = -2 sin[(θ+φ)/2] sin[(θ-φ)/2]
tanA+tanB=sin(A+B)/cosAcosB=tan(A+B)(1-tanAtanB)
tanA-tanB=sin(A-B)/cosAcosB=tan(A-B)(1+tanAtanB)
积化和差
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
诱导公式
sin(-α) = -sinα
cos(-α) = cosα
tan (—a)=-tanα
sin(π/2-α) = cosα
cos(π/2-α) = sinα
sin(π/2+α) = cosα
cos(π/2+α) = -sinα
sin(π-α) = sinα
cos(π-α) = -cosα
sin(π+α) = -sinα
cos(π+α) = -cosα
tanA= sinA/cosA
tan(π/2+α)=-cotα
tan(π/2-α)=cotα
tan(π-α)=-tanα
tan(π+α)=tanα
诱导公式记背诀窍:奇变偶不变,符号看象限
万能公式
sinα=2tan(α/2)/[1+tan^(α/2)]
cosα=[1-tan^(α/2)]/1+tan^(α/2)]
tanα=2tan(α/2)/[1-tan^(α/2)]
其它公式
(1)(sinα)^2+(cosα)^2=1
(2)1+(tanα)^2=(secα)^2
(3)1+(cotα)^2=(cscα)^2
证明下面两式,只需将一式,左右同除(sinα)^2,第二个除(cosα)^2即可
(4)对于任意非直角三角形,总有
tanA+tanB+tanC=tanAtanBtanC
证:
A+B=π-C
tan(A+B)=tan(π-C)
(tanA+tanB)/(1-tanAtanB)=(tanπ-tanC)/(1+tanπtanC)
整理可得
tanA+tanB+tanC=tanAtanBtanC
得证
同样可以得证,当x+y+z=nπ(n∈Z)时,该关系式也成立
由tanA+tanB+tanC=tanAtanBtanC可得出以下结论
(5)cotAcotB+cotAcotC+cotBcotC=1
(6)cot(A/2)+cot(B/2)+cot(C/2)=cot(A/2)cot(B/2)cot(C/2)
(7)(cosA)^2+(cosB)^2+(cosC)^2=1-2cosAcosBcosC
(8)(sinA)^2+(sinB)^2+(sinC)^2=2+2cosAcosBcosC
(9)sinα+sin(α+2π/n)+sin(α+2π*2/n)+sin(α+2π*3/n)+……+sin[α+2π*(n-1)/n]=0
cosα+cos(α+2π/n)+cos(α+2π*2/n)+cos(α+2π*3/n)+……+cos[α+2π*(n-1)/n]=0 以及
sin^2(α)+sin^2(α-2π/3)+sin^2(α+2π/3)=3/2
tanAtanBtan(A+B)+tanA+tanB-tan(A+B)=0
这题目我居然做过
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绛旓細1銆佸叕寮忎竴锛氫换鎰忚伪涓-伪鐨涓夎鍑芥暟鍊间箣闂寸殑鍏崇郴锛歴in锛堬紞伪锛=锛峴in伪cos锛堬紞伪锛=cos伪tan锛堬紞伪锛=锛峵an伪cot锛堬紞伪锛=锛峜ot伪 2銆佸叕寮忎簩锛歴in锛埾+伪锛=鈥攕in伪cos锛埾+伪锛=锛峜os伪tan锛埾+伪锛=tan伪cot锛埾+伪锛=cot伪 3銆佸叕寮忎笁锛氬埄鐢ㄥ叕寮忎簩鍜屽叕寮忎笁鍙互寰楀埌蟺-伪涓幬辩殑涓...
绛旓細楂樹腑涓夎鍑芥暟鐨勫叏閮ㄥ叕寮忔暣鐞嗗涓嬶細涓銆侀珮涓笁瑙掑嚱鏁扮殑鍏ㄩ儴鍏紡 1銆佸拰宸鍏紡锛歴in(A+B)=sinAcosB+cosAsinB锛宻in(A-B)=sinAcosB-cosAsinB锛宑os(A+B)=cosAcosB-sinAsinB锛宑os(A-B)=cosAcosB+sinAsinB銆2銆佺Н鍖栧拰宸叕寮忥細sinAcosB=1/2*(sin(A+B)+sin(A-B))锛宑osAsinB=1/2*(sin(A+B)...
绛旓細楂樹腑鏁板 涓夎鍑芥暟鍏紡澶у叏 涓夎鍑芥暟杩欑珷鍏紡寰堝锛屽挨鍏舵槸璇卞鍏紡灏辨湁浜屽崄澶氫釜锛屽叏閮ㄨ蹇嗘槸姣旇緝鍚冨姏銆傚熀纭寮辩殑鍚屽鏇村簲璇ュソ濂借璁拌繖浜涘叕寮忥紝鐔熺粌杩欎簺鍏紡涔熷氨鎶撲綇浜嗚繖绔犵殑閲嶇偣浜嗐傚涔犺捣鏉ヤ簨鍗婂姛鍊嶃備袱瑙掑拰 sin(A+B)=sinAcosB+cosAsinB sin(A-B)=sinAcosB-sinBcosA cos(A+B)=cosAcosB-sinAsinB ...
绛旓細涓銆佷换鎰忚鐨涓夎鍑芥暟 1锛庝笁瑙掑嚱鏁扮殑瀹氫箟锛氳 鏄竴涓换鎰忚锛岀偣 鏄 鐨勭粓杈逛笌鍗曚綅鍦嗙殑浜ょ偣锛岄偅涔堬細 鍙仛 鐨勬寮︼紝璁颁綔 锛屽嵆 锛 鍙仛 鐨勪綑寮︼紝璁颁綔 锛屽嵆 锛涘彨鍋 鐨勬鍒囷紝璁颁綔 锛屽嵆 .姝e鸡銆佷綑寮︺佹鍒囬兘鏄互瑙掍负鑷彉閲忥紝浠ュ崟浣嶅渾涓婄偣鐨勫潗鏍囨垨鍧愭爣鐨勬瘮鍊间负鍑芥暟鍊肩殑鍑芥暟锛屾垜浠皢瀹冧滑缁熺О涓轰笁瑙...
绛旓細浣欏鸡鍑芥暟 cos胃=x/r 姝e垏鍑芥暟 tan胃=y/x 浣欏垏鍑芥暟 cot胃=x/y 姝e壊鍑芥暟 sec胃=r/x 浣欏壊鍑芥暟 csc胃=r/y 浠ュ強涓や釜涓嶅父鐢紝宸茶秼浜庤娣樻卑鐨勫嚱鏁帮細姝g煝鍑芥暟 versin胃 =1-cos胃 浣欑煝鍑芥暟 vercos胃 =1-sin胃 鍚岃涓夎鍑芥暟闂寸殑鍩烘湰鍏崇郴寮忥細路骞虫柟鍏崇郴锛歴in^2(伪)+cos^2(伪)=1 tan^2(伪...
绛旓細楂樹腑甯哥敤鐨涓夎鍑芥暟鍊艰〃閫氬父鍖呮嫭姝e鸡鍑芥暟锛坰in锛夈佷綑寮﹀嚱鏁帮紙cos锛夈佹鍒囧嚱鏁帮紙tan锛夈佷綑鍒囧嚱鏁帮紙cot锛夈佹鍓插嚱鏁帮紙sec锛夊拰浣欏壊鍑芥暟锛坈sc锛夊湪鐗瑰畾瑙掑害涓婄殑鏁板笺傝繖浜涘嚱鏁板艰〃鎻愪緵浜嗙壒瀹氳搴︾殑涓夎鍑芥暟鍙栧硷紝浣垮緱瀛︾敓浠彲浠ュ湪瑙i鎴栬绠楄繃绋嬩腑蹇熸煡鎵惧弬鑰冦傚父瑙佺殑涓夎鍑芥暟鍊艰〃涓鑸粰鍑轰簡0搴﹀埌360搴︼紙鎴0鍒...
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