三角函数

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})