admin 发表于 2024-9-6 10:06:46

初中数学|三角函数知识点&公式全归纳

初中数学,让学生头痛的很大一部分就是三角函数!很多同学对与三角函数中正弦、余弦、正切、余切中的公式容易混淆,做题的时候不能够运用正确的公式,以至于三角函数题成为了他们失分的重要部分。

最近也有不少童鞋在微信上问三角函数这一块。为了帮助同学们能够熟练掌握这一部分知识,今天就来介绍下三角函数以及归纳一些知识点。



锐角三角函数定义

锐角角A的正弦(sin),余弦(cos)和正切(tan),余切(cot)以及正割(sec),余割(csc)都叫做角A的锐角三角函数。

正弦(sin):对边比斜边,即sinA=a/c

余弦(cos):邻边比斜边,即cosA=b/c

正切(tan):对边比邻边,即tanA=a/b

余切(cot):邻边比对边,即cotA=b/a

正割(sec):斜边比邻边,即secA=c/b

余割(csc):斜边比对边,即cscA=c/a



特殊角三角函数值

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_0.png



三角函数关系

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_1.png

互余角的关系

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_2.png

sin(90°-α)=cosα, cos(90°-α)=sinα,

tan(90°-α)=cotα, cot(90°-α)=tanα.

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_3.png

平方关系

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_4.png

sin^2(α)+cos^2(α)=1

tan^2(α)+1=sec^2(α)

cot^2(α)+1=csc^2(α)

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_5.png

积的关系

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_6.png

sinα=tanα·cosα

cosα=cotα·sinα

tanα=sinα·secα

cotα=cosα·cscα

secα=tanα·cscα

cscα=secα·cotα

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_7.png

倒数关系

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_8.png

tanα·cotα=1

sinα·cscα=1

cosα·secα=1



锐角三角函数公式

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_9.png

两角和差公式

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_10.png

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)

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_11.png

三角和的公式

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_12.png

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)

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_13.png

倍角公式

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_14.png

tan2A = 2tanA/(1-tan² A)

Sin2A=2SinA•CosA

Cos2A = Cos^2 A--Sin² A =2Cos² A-1 =1-2sin^2 A

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_15.png

三倍角公式

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_16.png

sin3A = 3sinA-4(sinA)³;

cos3A = 4(cosA)³ -3cosA

tan3a = tan a • tan(π/3+a)• tan(π/3-a)

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_17.png

半角公式

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_18.png

sin(α/2)=±√((1-cosα)/2)

cos(α/2)=±√((1+cosα)/2)

tan(α/2)=±√((1-cosα)/(1+cosα))=sinα/(1+cosα)=(1-cosα)/sinα

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_19.png

积化和差公式

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_20.png

sinα·cosβ=(1/2)

cosα·sinβ=(1/2)

cosα·cosβ=(1/2)

sinα·sinβ=-(1/2)

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_21.png

和差化积公式

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_22.png

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]

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_23.png

万能公式

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_24.png

sin(a) = / {1+²}

cos(a) = {1-^2} / {1+²}

tan(a) = /{1-^2}

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_25.png

推导公式

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_26.png

tanα+cotα=2/sin2α

tanα-cotα=-2cot2α

1+cos2α=2cos^2α

1-cos2α=2sin^2α

1+sinα=(sinα/2+cosα/2)^2



三角形面积定理



三角函数的图象性质

http://kuailexuexi.net/data/attachment/forum/20240906/1725588406828_27.jpg
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