三角函數(shù)公式

2022-07-20 21:27 作者:愛(ài)喝殺蟲劑的Isaac 0人讀過(guò) | 我要投稿

三角恒等式

$sin%5E2x%2Bcos%5E2x%3D1$

$%5Cbegin%7Barray%7D%7Bl%7D%0A%5Ctan%20%5E%7B2%7D%20x%2B1%3D%5Csec%20%5E%7B2%7D%20x%20%5C%5C%0A1%2B%5Ccot%20%5E%7B2%7D%20x%3D%5Ccsc%20%5E%7B2%7D%20x%0A%5Cend%7Barray%7D$

和差公式

$%5Csin%20(x%2By)%3D%5Csin%20x%20%5Ccos%20y%2B%5Ccos%20x%20%5Csin%20y$

$%5Csin%20(x-y)%3D%5Csin%20x%20%5Ccos%20y-%5Ccos%20x%20%5Csin%20y$

$%5Ccos%20(x%2By)%3D%5Ccos%20x%20%5Ccos%20y-%5Csin%20x%20%5Csin%20y$

$%5Ccos%20(x-y)%3D%5Ccos%20x%20%5Ccos%20y%2B%5Csin%20x%20%5Csin%20y$

$%5Ctan%20(x%2By)%3D%5Cfrac%7B%5Ctan%20x%2B%5Ctan%20y%7D%7B1-%5Ctan%20x%20%5Ctan%20y%7D$

$%5Ctan%20(x-y)%3D%5Cfrac%7B%5Ctan%20x-%5Ctan%20y%7D%7B1%2B%5Ctan%20x%20%5Ctan%20y%7D$

二倍角公式

$%5Csin%20(2%20x)%3D2%20%5Csin%20x%20%5Ccos%20x$

$%5Ccos%20(2%20x)%3D%5Ccos%20%5E%7B2%7D%20x-%5Csin%20%5E%7B2%7D%20x$

$%5Ctan%20(2%20x)%3D%5Cfrac%7B2%20%5Ctan%20x%7D%7B1-%5Ctan%20%5E%7B2%7D%20x%7D$

半角公式

$%5Csin%20%5Cfrac%7Bx%7D%7B2%7D%3D%5Cpm%20%5Csqrt%7B%5Cfrac%7B1-%5Ccos%20x%7D%7B2%7D%7D$

$%5Ccos%20%5Cfrac%7Bx%7D%7B2%7D%3D%5Cpm%20%5Csqrt%7B%5Cfrac%7B1%2B%5Ccos%20x%7D%7B2%7D%7D$

$%5Ctan%20%5Cfrac%7Bx%7D%7B2%7D%3D%5Cpm%20%5Csqrt%7B%5Cfrac%7B1-%5Ccos%20x%7D%7B1%2B%5Ccos%20x%7D%7D%3D%5Cfrac%7B1-%5Ccos%20x%7D%7B%5Csin%20x%7D%3D%5Cfrac%7B%5Csin%20x%7D%7B1%2B%5Ccos%20x%7D$

級(jí)數(shù)定義

$%5Csin%20x%3D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B(-1)%5E%7Bn%7D%20x%5E%7B2%20n%2B1%7D%7D%7B(2%20n%2B1)%20!%7D%3Dx-%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%20!%7D%2B%5Cfrac%7Bx%5E%7B5%7D%7D%7B5%20!%7D-%5Cfrac%7Bx%5E%7B7%7D%7D%7B7%20!%7D%2B%5Ccdots$

$%5Ccos%20x%3D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B(-1)%5E%7Bn%7D%20x%5E%7B2%20n%7D%7D%7B(2%20n)%20!%7D%3D1-%5Cfrac%7Bx%5E%7B2%7D%7D%7B2%20!%7D%2B%5Cfrac%7Bx%5E%7B4%7D%7D%7B4%20!%7D-%5Cfrac%7Bx%5E%7B6%7D%7D%7B6%20!%7D%2B%5Ccdots$

$%5Ctan%20x%3D%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B(-1)%5E%7Bn-1%7D%202%5E%7B2%20n%7D%5Cleft(2%5E%7B2%20n%7D-1%5Cright)%20B_%7B2%20n%7D%20x%5E%7B2%20n-1%7D%7D%7B(2%20n)%20!%7D%3Dx%2B%5Cfrac%7Bx%5E%7B3%7D%7D%7B3%7D%2B%5Cfrac%7B2%20x%5E%7B5%7D%7D%7B15%7D%2B%5Cfrac%7B17%20x%5E%7B7%7D%7D%7B315%7D%2B%5Ccdots%5Cleft(%7Cx%7C%3C%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)$

$%5Ccsc%20x%3D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B(-1)%5E%7Bn%2B1%7D%202%5Cleft(2%5E%7B2%20n-1%7D-1%5Cright)%20B_%7B2%20n%7D%20x%5E%7B2%20n-1%7D%7D%7B(2%20n)%20!%7D%3D%5Cfrac%7B1%7D%7Bx%7D%2B%5Cfrac%7Bx%7D%7B6%7D%2B%5Cfrac%7B7%20x%5E%7B3%7D%7D%7B360%7D%2B%5Cfrac%7B31%20x%5E%7B5%7D%7D%7B15120%7D%2B%5Ccdots(0%3C%7Cx%7C%3C%5Cpi)$

$%5Csec%20x%3D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B(-1)%5E%7Bn%7D%20E_%7Bn%7D%20x%5E%7B2%20n%7D%7D%7B(2%20n)%20!%7D%3D1%2B%5Cfrac%7Bx%5E%7B2%7D%7D%7B2%7D%2B%5Cfrac%7B5%20x%5E%7B4%7D%7D%7B24%7D%2B%5Cfrac%7B61%20x%5E%7B6%7D%7D%7B720%7D%2B%5Ccdots%5Cleft(%7Cx%7C%3C%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright)$

$%5Ccot%20x%3D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B(-1)%5E%7Bn%7D%202%5E%7B2%20n%7D%20B_%7B2%20n%7D%20x%5E%7B2%20n-1%7D%7D%7B(2%20n)%20!%7D%3D%5Cfrac%7B1%7D%7Bx%7D-%5Cfrac%7Bx%7D%7B3%7D-%5Cfrac%7Bx%5E%7B3%7D%7D%7B45%7D-%5Cfrac%7B2%20x%5E%7B5%7D%7D%7B945%7D-%5Ccdots(0%3C%7Cx%7C%3C%5Cpi)$

正弦定理

$%5Cfrac%7Ba%7D%7B%5Csin%20A%7D%3D%5Cfrac%7Bb%7D%7B%5Csin%20B%7D%3D%5Cfrac%7Bc%7D%7B%5Csin%20C%7D%3D2%20R$

余弦定理

$c%5E%7B2%7D%3Da%5E%7B2%7D%2Bb%5E%7B2%7D-2%20a%20b%20%5Ccos%20C$

$%5Ccos%20C%3D%5Cfrac%7Ba%5E%7B2%7D%2Bb%5E%7B2%7D-c%5E%7B2%7D%7D%7B2%20a%20b%7D$

正切定理

$%5Cfrac%7Ba%2Bb%7D%7Ba-b%7D%3D%5Cfrac%7B%5Ctan%20%5Cfrac%7BA%2BB%7D%7B2%7D%7D%7B%5Ctan%20%5Cfrac%7BA-B%7D%7B2%7D%7D$

余切定理

$%20%5Ccot%20%5Cfrac%7B%5Calpha%7D%7B2%7D%3D%5Cfrac%7Bs-a%7D%7B%5Czeta%7D%20$

$%5Ctext%20%7B%20%E5%85%B6%E4%B8%AD%20%7D%20%5Czeta%3D%5Csqrt%7B%5Cfrac%7B1%7D%7Bs%7D(s-a)(s-b)(s-c)%7D%20%5Ctext%20%7B%20%E4%B8%BA%E4%B8%89%E8%A7%92%E5%BD%A2%E7%9A%84%E5%86%85%E5%88%87%E5%9C%86%E5%8D%8A%E5%BE%84%EF%BC%8C%20%7D%20s%3D%5Cfrac%7Ba%2Bb%2Bc%7D%7B2%7D%20%5Ctext%20%7B%20%E4%B8%BA%E4%B8%89%E8%A7%92%E5%BD%A2%E7%9A%84%E5%8D%8A%E5%91%A8%E9%95%BF%20%7D$

微積分

$%5Cbegin%7Barray%7D%7B%7Cl%7Cc%7Cc%7Cc%7Cc%7Cc%7Cc%7C%7D%0A%5Chline%20%5Ctext%20%7B%20%E5%87%BD%E6%95%B0%20%7D%20%26%20%5Csin%20x%20%26%20%5Ccos%20x%20%26%20%5Ctan%20x%20%26%20%5Ccot%20x%20%26%20%5Csec%20x%20%26%20%5Ccsc%20x%20%5C%5C%0A%5Chline%20%5Ctext%20%7B%20%E5%AF%BC%E5%87%BD%E6%95%B0%20%7D%20%26%20%5Ccos%20x%20%26%20-%5Csin%20x%20%26%20%5Csec%20%5E%7B2%7D%20x%20%26%20-%5Ccsc%20%5E%7B2%7D%20x%20%26%20%5Csec%20x%20%5Ctan%20x%20%26%20-%5Ccsc%20x%20%5Ccot%20x%20%5C%5C%0A%5Chline%20%5Ctext%20%7B%20%E5%8F%8D%E5%AF%BC%E5%87%BD%E6%95%B0%20%7D%20%26%20-%5Ccos%20x%20%26%20%5Csin%20x%20%26%20-%5Cln%20%7C%5Ccos%20x%7C%20%26%20%5Cln%20%7C%5Csin%20x%7C%20%26%20%5Cln%20%7C%5Csec%20x%2B%5Ctan%20x%7C%20%26%20%5Cln%20%7C%5Ccsc%20x-%5Ccot%20x%7C%20%5C%5C%0A%5Chline%0A%5Cend%7Barray%7D$

三角函數(shù)的特殊值

$%5Cbegin%7Barray%7D%7B%7Cc%7Cc%7Cc%7Cc%7Cc%7Cc%7Cc%7Cc%7C%7D%0A%5Chline%20%5Ctext%20%7B%20%E5%87%BD%E6%95%B0%E5%90%8D%20%7D%20%26%200%5Cleft(0%5E%7B%5Ccirc%7D%5Cright)%20%26%20%5Cfrac%7B%5Cpi%7D%7B12%7D%5Cleft(15%5E%7B%5Ccirc%7D%5Cright)%20%26%20%5Cfrac%7B%5Cpi%7D%7B6%7D%5Cleft(30%5E%7B%5Ccirc%7D%5Cright)%20%26%20%5Cfrac%7B%5Cpi%7D%7B4%7D%5Cleft(45%5E%7B%5Ccirc%7D%5Cright)%20%26%20%5Cfrac%7B%5Cpi%7D%7B3%7D%5Cleft(60%5E%7B%5Ccirc%7D%5Cright)%20%26%20%5Cfrac%7B5%20%5Cpi%7D%7B12%7D%5Cleft(75%5E%7B%5Ccirc%7D%5Cright)%20%26%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cleft(90%5E%7B%5Ccirc%7D%5Cright)%20%5C%5C%0A%5Chline%20%5Csin%20%26%200%20%26%20%5Cfrac%7B%5Csqrt%7B6%7D-%5Csqrt%7B2%7D%7D%7B4%7D%20%26%20%5Cfrac%7B1%7D%7B2%7D%20%26%20%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%20%26%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%20%26%20%5Cfrac%7B%5Csqrt%7B6%7D%2B%5Csqrt%7B2%7D%7D%7B4%7D%20%26%201%20%5C%5C%0A%5Chline%20%5Ccos%20%26%201%20%26%20%5Cfrac%7B%5Csqrt%7B6%7D%2B%5Csqrt%7B2%7D%7D%7B4%7D%20%26%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B2%7D%20%26%20%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%20%26%20%5Cfrac%7B1%7D%7B2%7D%20%26%20%5Cfrac%7B%5Csqrt%7B6%7D-%5Csqrt%7B2%7D%7D%7B4%7D%20%26%200%20%5C%5C%0A%5Chline%20%5Ctan%20%26%200%20%26%202-%5Csqrt%7B3%7D%20%26%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B3%7D%20%26%201%20%26%20%5Csqrt%7B3%7D%20%26%202%2B%5Csqrt%7B3%7D%20%26%20%5Cpm%20%5Cinfty%20%5C%5C%0A%5Chline%20%5Ccot%20%26%20%5Cpm%20%5Cinfty%20%26%202%2B%5Csqrt%7B3%7D%20%26%20%5Csqrt%7B3%7D%20%26%201%20%26%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B3%7D%20%26%202-%5Csqrt%7B3%7D%20%26%200%20%5C%5C%0A%5Chline%20%5Csec%20%26%201%20%26%20%5Csqrt%7B6%7D-%5Csqrt%7B2%7D%20%26%20%5Cfrac%7B2%20%5Csqrt%7B3%7D%7D%7B3%7D%20%26%20%5Csqrt%7B2%7D%20%26%202%20%26%20%5Csqrt%7B6%7D%2B%5Csqrt%7B2%7D%20%26%20%5Cpm%20%5Cinfty%20%5C%5C%0A%5Chline%20%5Ccsc%20%26%20%5Cpm%20%5Cinfty%20%26%20%5Csqrt%7B6%7D%2B%5Csqrt%7B2%7D%20%26%202%20%26%20%5Csqrt%7B2%7D%20%26%20%5Cfrac%7B2%20%5Csqrt%7B3%7D%7D%7B3%7D%20%26%20%5Csqrt%7B6%7D-%5Csqrt%7B2%7D%20%26%201%20%5C%5C%0A%5Chline%0A%5Cend%7Barray%7D$

反三角函數(shù)

$%5Cbegin%7Barray%7D%7B%7Cc%7Cc%7Cc%7C%7D%0A%5Chline%20%5Ctext%20%7B%20%E5%8F%8D%E4%B8%89%E8%A7%92%E5%87%BD%E6%95%B0%20%7D%20%26%20%5Ctext%20%7B%20%E5%AE%9A%E4%B9%89%20%7D%20%26%20%5Ctext%20%7B%20%E5%80%BC%E5%9F%9F%20%7D%20%5C%5C%0A%5Chline%20%5Carcsin%20(x)%3Dy%20%26%20%5Csin%20(y)%3Dx%20%26%20-%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cleq%20y%20%5Cleq%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5C%5C%0A%5Chline%20%5Carccos%20(x)%3Dy%20%26%20%5Ccos%20(y)%3Dx%20%26%200%20%5Cleq%20y%20%5Cleq%20%5Cpi%20%5C%5C%0A%5Chline%20%5Carctan%20(x)%3Dy%20%26%20%5Ctan%20(y)%3Dx%20%26%20-%5Cfrac%7B%5Cpi%7D%7B2%7D%3Cy%3C%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5C%5C%0A%5Chline%20%5Coperatorname%7Barccsc%7D(x)%3Dy%20%26%20%5Ccsc%20(y)%3Dx%20%26%20-%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5Cleq%20y%20%5Cleq%20%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%20y%20%5Cneq%200%20%5C%5C%0A%5Chline%20%5Coperatorname%7Barcsec%7D(x)%3Dy%20%26%20%5Csec%20(y)%3Dx%20%26%200%20%5Cleq%20y%20%5Cleq%20%5Cpi%2C%20y%20%5Cneq%20%5Cfrac%7B%5Cpi%7D%7B2%7D%20%5C%5C%0A%5Chline%20%5Coperatorname%7Barccot%7D(x)%3Dy%20%26%20%5Ccot%20(y)%3Dx%20%26%200%3Cy%3C%5Cpi%20%5C%5C%0A%5Chline%0A%5Cend%7Barray%7D$

標(biāo)簽：