三角函数和反三角函数公式

一．三角函数公式

1.诱导公式

sin(-a) = - sin(a)

cos(-a) = cos(a)

sin(π/2（90度） - a) = cos(a)

cos(π/2（90度） - a) = sin(a)

sin(π/2 （90度）+ a) = cos(a)

cos(π/2 （90度）+ a) = - sin(a)

sin(π（180度）- a) = sin(a)

cos(π（180度） - a) = - cos(a)

sin(π（180度）+ a) = - sin(a)

cos(π（180度）+ 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 + 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]

4.积化和差公式

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)]

5.二倍角公式

sin(2a) = 2sin(a)cos(b)

cos(2a) = cos2(a) - sin2(a) = 2cos2(a) -1=1 - 2sin2(a)

6.半角公式

sin2(a/2) = [1 - cos(a)] / 2

cos2(a/2) = [1 + cos(a)] / 2

tan(a/2) = [1 - cos(a)] /sin(a) = sina / [1 + cos(a)]

7.万能公式

sin(a) = 2tan(a/2) / [1+tan2(a/2)]

cos(a) = [1-tan2(a/2)] / [1+tan2(a/2)]

tan(a) = 2tan(a/2) / [1-tan2(a/2)

二．反三角函数公式

反三角函数其他公式：

cos(arcsinx)=√(1-x^2)

arcsin(-x)=-arcsinx

arccos(-x)=π-arccosx

arctan(-x)=-arctanx

arccot(-x)=π-arccotx

arcsinx+arccosx=π/2=arctanx+arccotx

sin(arcsinx)=cos(arccosx)=tan(arctanx)=cot(arccotx)=x

arcsin x = x + x^3/(2*3) + (1*3)x^5/(2*4*5) +

1*3*5(x^7)/(2*4*6*7）……+(2k+1)!!*x^(2k-1)/(2k!!*(2k+1))+……（|x|<1) !！表示双阶乘

arccos x = π -(x + x^3/(2*3) + (1*3)x^5/(2*4*5) + 1*3*5(x^7)/(2*4*6*7）……）（|x|<1)

arctan x = x - x^3/3 + x^5/5 -……

举例

当x∈[-π/2，π/2] 有arcsin(sinx)=x

x∈[0，π]，arccos(cosx)=x

x∈（-π/2，π/2），arctan(tanx)=x

x∈（0，π），arccot(cotx)=x

x>0，arctanx=π/2-arctan1/x，arccotx类似

若(arctanx+arctany）∈（-π/2，π/2），则arctanx+arctany=arctan((x+y)/(1-xy)) 例如，arcsinχ表示角α，满足α∈[-π/2，π/2]且sinα=χ;arccos(-4/5)表示角β，满足β∈[0，π]且cosβ=-4/5;arctan2表示角φ，满足φ∈（-π/2，π/2)且tanφ=2