特别为了某人写的23333,可能会有点问题23333
| ( heta) | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° |
|---|---|---|---|---|---|---|---|---|---|
| 弧度 | 0 | (frac{pi}{6}) | (frac{pi}{4}) | (frac{pi}{3}) | (frac{pi}{2}) | (frac{2pi}{3}) | (frac{3pi}{4}) | (frac{5pi}{6}) | (pi) |
| (sin{ heta}) | 0 | (frac{1}{2}) | (frac{sqrt{2}}{2}) | (frac{sqrt{3}}{2}) | 1 | (frac{sqrt{3}}{2}) | (frac{sqrt{2}}{2}) | (frac{1}{2}) | 0 |
| (cos{ heta}) | 1 | (frac{sqrt{3}}{2}) | (frac{sqrt{2}}{2}) | (frac{1}{2}) | 0 | -(frac{1}{2}) | -(frac{sqrt{2}}{2}) | -(frac{sqrt{3}}{2}) | -1 |
| ( an{ heta}) | 0 | (frac{sqrt{3}}{3}) | 1 | (sqrt{3}) | 无 | -(sqrt{3}) | -1 | -(frac{sqrt{3}}{3}) | 0 |
| ( heta) | 210° | 225° | 240° | 270° | 300° | 315° | 330° | 360° | 0° |
|---|---|---|---|---|---|---|---|---|---|
| 弧度 | (frac{7pi}{6}) | (frac{5pi}{4}) | (frac{4pi}{3}) | (frac{3pi}{2}) | (frac{5pi}{3}) | (frac{7pi}{4}) | (frac{11pi}{6}) | 0 | 0 |
| (sin{ heta}) | -(frac{1}{2}) | -(frac{sqrt{2}}{2}) | -(frac{sqrt{3}}{2}) | -1 | -(frac{sqrt{3}}{2}) | -(frac{sqrt{2}}{2}) | -(frac{1}{2}) | 0 | 0 |
| (cos{ heta}) | -(frac{sqrt{3}}{2}) | -(frac{sqrt{2}}{2}) | -(frac{1}{2}) | 0 | (frac{1}{2}) | (frac{sqrt{2}}{2}) | (frac{sqrt{3}}{2}) | 1 | 1 |
| ( an{ heta}) | (frac{sqrt{3}}{3}) | 1 | (sqrt{3}) | 无 | -(sqrt{3}) | -1 | -(frac{sqrt{3}}{3}) | 0 | 0 |
基本公式:
(sin^2{alpha}+cos^2{alpha}=1)
(frac{sin{alpha}}{cos{alpha}}= an{alpha})
诱导公式:
(sin{(alpha+k imes2pi)}=sin{alpha} quad cos{(alpha+k imes2pi)}=cos{alpha} quad an{(alpha+k imes2pi)}= an{alpha} quad (k in Z))
(sin{(alpha+pi)}=-sin{alpha} quad cos{(alpha+pi)}=-cos{alpha} quad an{(alpha+pi)}= an{alpha})
(sin{(-alpha)}=-sin{alpha} quad cos{(-alpha)}=cos{alpha} quad an{(-alpha)}=- an{alpha})
(sin{(pi-alpha)}=sin{alpha} quad cos{(pi-alpha)}=-cos{alpha} quad an{(pi-alpha)}=- an{alpha})
$sin{(frac{pi}{2}-alpha)}=cos{alpha} quad cos{(frac{pi}{2}-alpha)}=sin{alpha} $
$sin{(frac{pi}{2}+alpha)}=cos{alpha} quad cos{(frac{pi}{2}+alpha)}=-sin{alpha} $
口诀:
奇变偶不变,符号看象限
加减法:
(sin{(alpha+eta)}=sin{alpha} imescos{eta}+sin{eta} imescos{alpha}quadsin{(alpha-eta)}=sin{alpha} imescos{eta}-sin{eta} imescos{alpha})
(cos{(alpha+eta)}=cos{alpha} imescos{eta}-sin{eta} imessin{alpha}quadcos{(alpha-eta)}=cos{alpha} imescos{eta}+sin{eta} imessin{alpha})
( an{(alpha+eta)}=frac{ an{alpha}+ an{eta}}{1- an{alpha} imes an{eta}}quad an{(alpha-eta)}=frac{ an{alpha}- an{eta}}{1+ an{alpha} imes an{eta}})
二倍角公式:
(sin{(2 imesalpha)}=2 imessin{alpha} imescos{alpha})
(cos{(2 imesalpha)}=cos^2{alpha}-sin^2{alpha}=2 imescos^2{alpha}-1=1-2 imessin^2{alpha})
( an{(2 imesalpha)}=frac{2 imes an{alpha}}{1- an^2{alpha}})
( an{frac{alpha}{2}}=frac{sin{alpha}}{1+cos{alpha}}=frac{1-cos{alpha}}{sin{alpha}})
(sin^2{alpha}=frac{1-cos{(2 imesalpha)}}{2}quadcos^2{alpha}=frac{sin{(2 imesalpha)}-1}{2})
((sin{alpha}+cos{alpha})^2=1+sin{2 imesalpha})
和差化积:
(sin{alpha}-sin{eta}=2 imescos{frac{alpha+eta}{2}} imessin{frac{alpha-eta}{2}}quadsin{alpha}+sin{eta}=2 imescos{frac{alpha-eta}{2}} imessin{frac{alpha+eta}{2}})
(cos{alpha}-cos{eta}=2 imescos{frac{alpha+eta}{2}} imescos{frac{alpha-eta}{2}}quadcos{alpha}+cos{eta}=2 imessin{frac{alpha-eta}{2}} imessin{frac{alpha+eta}{2}})
积化和差:
(sin{alpha} imescos{eta}=frac{sin{(alpha+eta)}+sin{(alpha-eta)}}{2}quadsin{alpha} imessin{eta}=frac{cos{(alpha+eta)}-cos{(alpha-eta)}}{2})
(cos{alpha} imessin{eta}=frac{sin{(alpha+eta)}-sin{(alpha-eta)}}{2}quadcos{alpha} imescos{eta}=frac{cos{(alpha+eta)}+cos{(alpha-eta)}}{2})
在任意三角形ABC中,定义角A对边为a,角B对边为b,角C对边为c,则有:
1.(sin{(A+B)}=sin{C}quadcos{(A+B)}=-cos{C})
2.正弦定理:
(frac{a}{sin{A}}=frac{b}{sin{B}}=frac{c}{sin{C}}=2R),其中R为该三角形外接圆的半径
3.余弦定理:
(c^2=a^2+b^2-2 imes a imes b imes cos{C} quad a^2=b^2+c^2-2 imes b imes c imes cos{A} quad b^2=a^2+c^2-2 imes a imes c imes cos{B})
对于三角函数: (f(x)=Asin{(omega x+varphi)})
(A): 振幅
$ omega $: 三角函数在y轴方向的压缩程度,当 $ omega > 1$ 时,表示被压缩, (omega < 1) 时表示拉伸.
(omega x+varphi) : 三角函数的相位
(varphi) : 三角函数的初相
求周期(T): (T= frac{2 pi}{omega})
求频率(f): (f=frac{1}{T})