Реферат: Математика (Шпаргалка)

sinиcosсуммы и разности двух аргументов

sin(a

±b)=sin a·cosb±sinb·cosa

cos(a

±b)=cosa·cosb`+sin a·sinb

tg a

± tg b

tg (a

±b) = 1 ±tg a· tg b

tg (a

±b) =

= ctg a

·ctg b`+ 1 = 1 ±tg a· tg b

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Тригонометрические функциидвойногоаргумента

sin2x=2sinx cosx

cos 2x = cos2x — sin2x=

= 2cos2x-1=1-2sin2x

tg2x= 2 tgx

1 - tg2x

sin 3x =3sin x — 4 sin3x

cos 3x= 4 cos3 x — 3 cos

<img src="/cache/referats/2961/image001.gif" v:shapes="_x0000_s1027"><img src="/cache/referats/2961/image002.gif" v:shapes="_x0000_s1026">ВАЖНО: знак перед корнем зависит от того, где нах-ся угол ½ x:

<img src="/cache/referats/2961/image003.gif" v:shapes="_x0000_s1028">sin ½ x= ±

1-cosx

<img src="/cache/referats/2961/image004.gif" v:shapes="_x0000_s1030"><img src="/cache/referats/2961/image005.gif" v:shapes="_x0000_s1029"> 2

<img src="/cache/referats/2961/image006.gif" v:shapes="_x0000_s1031">cos ½ x= ±

1+cosx

<img src="/cache/referats/2961/image007.gif" v:shapes="_x0000_s1032"><img src="/cache/referats/2961/image008.gif" v:shapes="_x0000_s1033">NB!Следующие формулы справедливыпри знаменателе ¹

0 и существования функций, входящих в эти формулы (tg, ctg)

<img src="/cache/referats/2961/image006.gif" v:shapes="_x0000_s1034">tg ½ x=sinx=1-cosx=±

1-cosx

1+cosx sinx 1+cosx

<img src="/cache/referats/2961/image008.gif" v:shapes="_x0000_s1036"><img src="/cache/referats/2961/image009.gif" v:shapes="_x0000_s1037"><img src="/cache/referats/2961/image007.gif" v:shapes="_x0000_s1035">сtg½ x=sinx=1+cosx=±

1+cosx

1-cosx sinx 1-cosx

Формулы понижения степени:

sin2 x = 1– cos 2x

cos2 x = 1+ cos 2x

sin3 x = 3 sin x – sin 3x

cos3 x = 3 cos x + cos 3x

Преобразование произведения двух функций в сумму:

2 sinxsiny = cos(x-y) – cos(x+y)

2cosx cosy = cos(x-y)+cos(x+y)

2sinx cosy = sin(x-y) + sin (x+y)

tgx tgy = tgx + tgy

ctgx+ ctgy

ctgx ctgy = ctgx + ctgy

tgx + tgy

tgx ctgy = tgx + ctgy

ctgx+ tgy

NB!Вышеперечисленные формулысправедливы при знаменателе ¹

0 и существования функций, входящих в эти формулы (tg, ctg)

sinx ±

siny= 2sin x±ycos x`+y

2 2

cosx + cosy =2cos x+ycos x-y

2 2

cosx — cosy = — 2sin x+ysin x-y

2 2

tgx ±

tgy= sin(x±y)

cosx cosy

tgx + сtgy = cos(x-y)

cosx siny

ctgx — tgy = cos(x+y)

sinx cosy

ctgx±

ctgy= sin(y±x)

sinx siny

sin x = 1 x=½p

+2pn, nÎZ

sin x= 0 x= p

n, nÎZ

sin x = -1 x=-½p

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a<span Courier New"; mso-ascii-font-family:«Times New Roman»;mso-hansi-font-family:«Times New Roman»; mso-bidi-font-family:«Times New Roman»;mso-ansi-language:EN-US;mso-char-type: symbol;mso-symbol-font-family:«Courier New»">]<span Courier New"; mso-ascii-font-family:«Times New Roman»;mso-hansi-font-family:«Times New Roman»; mso-bidi-font-family:«Times New Roman»;mso-ansi-language:EN-US;mso-char-type: symbol;mso-symbol-font-family:«Courier New»">≤1

x = (-1)karcsin a + p

k, kÎZ

cosx=1 x=2p

n, nÎZ

cosx=0 x=½p

+pn, nÎZ

cosx=-1 x=p

cosx=-½ x=±

2/3 p+2pn, nÎZ

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x=±

arccosa + 2pn, nÎZ

arccos(-x)=p

— arccos x

arcctg(-x)= p

— ctg x

tgx= 0 x= n, nÎ

ctgx= 0 x=½p

tg x= a x=arctg a +p

n, nÎZ

ctg x = a x=arcctga + p

n, nÎZ

Знаки тригонометрических функций в четвертях:

№f(a

)

sin

cos

ctg

-

-

-

III

-

-

a

рад=p×a°/180°; a°=a°×180°/p

Формулы ïðèâåäåíèÿ

– a

p

/2 ±a

p

±a

3/2 p

2p

– a

sin

-sin a

cos a

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+sin a

— cos a

— sin a

cos

cos a

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+sin a

— cos a

±

sin a

cos a

— tg a

+ ctg a

±

tg a

+ ctg a

— tg a

ctg