Àëãåáðà. Ãåîìåòðèÿ. Òðèãîíîìåòðèÿ (øïàðãàëêà)
Ôîðìóëû ñîêðàùåííîãî óìíîæåíèÿ
(à ± â)2 = à2 ± 2àâ + â2
(à ± â)3 = à3 ± 3à2â + 3àâ2 ± â3
à2 - â2 = (à + â) (à - â)
à3 + â3
= (à + â) (à2 - àâ + â2)
à3 - â3 = (à - â) (à2 + àâ + â2)
(à + â + ñ)2 = à2 + â2
+ ñ2 +2àâ +2àñ +2âñ
Ñòåïåíè.
àì àí
= àì + í
àì : àí = àì - í
(àâ)ì = àì
âì
(àì)í
= àìí
(à : â)ì = àì : âì
à- ì = 1 : àì
àì : í =
íÖ àì
Êîðíè.
íÖàâ =íÖà íÖâ
íÖà ìÖâ
= í ìÖàì âí
íÖà : â = íÖà : íÖâ
(íÖàì)õ = íÖàì õ
íÖàì = àì/í
ìÖíÖà
= ìíÖà
(íÖà)ì
= íÖàì
Àðèôìåòè÷åñêàÿ
ïðîãðåññèÿ.
à1, à2,
à3, …, à n-1, àn
à n-1
- àn = d
d – ðàçíîñòü
ïðîãðåññèè
à2 = à1+
d
à3 = à2
+ d = à1 + 2d
àn = à1 + d(n-1)
Sn = (à1 + àn) n = (2à1
+ ( n-1) d) n
2 2
Sn – ñóììà ÷ëåíîâ
àðèôìåòè÷åñêîé
ïðîãðåññèè.
d – ðàçíîñòü
ïðîãðåññèè.
d > 0 – ïðîãðåññèÿ
âîçðàñòàþùàÿ
d < 0 – ïðîãðåññèÿ
óáûâàþùàÿ.
Ãåîìåòðè÷åñêàÿ ïðîãðåññèÿ.
à1, à2,
à3, …, à n-1, àn
à n+1
/ àn
= q
à2 = à1
q
q - çíàìåíàòåëü ïðîãðåññèè.
à3 = à2
q = à1 q2
àn = à1 q n-1
Ñóììà ÷ëåíîâ äëÿ
âîçðàñòàþùåé
ïðîãðåññèè (q > 1)
Sn = àn q - à1 = à1 (qn
-1 : q – 1)
q – 1
Ñóììà ÷ëåíîâ äëÿ
óáûâàþùåé ïðîãðåññèè (q < 1)
Sn = à1 (1 - qn)
1 - q
Ñóììà ÷ëåíîâ
áåñêîíå÷íî óáûâàþùåé
Ïðîãðåññèè
Sn = à1
1 - q
Âåêòîðà.
à = Ì1Ì2
={õ2 – õ1, ó2 –
ó1, z2 –z1}
Äëèíà âåêòîðà
çà ç=Ö(õ2 - õ1)2 +(ó2
- ó1)2 + (z2 - z1)2
Óìíîæåíèå âåêòîðà íà ÷èñëî
a à = d
Ñêàëÿðíîå ïðîèçâåäåíèå âåêòîðîâ
à â = çà ççâ
çcos j
cos j = õ1õ2
+ ó1ó2 + z1z2
Öõ12 + ó12 +z12 Öõ22 +ó22 +
z22
à2 = çà ç2
à â = õ1õ2
+ ó1ó2 + z1z2
Ïàðàëëåëüíîñòü âåêòîðîâ
à ççâ, òî õ1 = ó1 = z1
õ2 ó2 z2
Ïåðïåíäèêóëÿðíîñòü
âåêòîðîâ
à ^ â, òî õ1õ2
+ ó1ó2 + z1z2
Ïðîèçâîäíàÿ.
(c u)¢ = ñ u¢
u ¢ = u¢ v – u v¢
v v2
(c)¢ = 0
(xn )¢ = n xn-1
(ax)¢ = ax
ln a
(åõ )¢ = åõ
(sin x)¢ = cos x
(cos x)¢ = - sin x
(tg x)¢ = 1
cos2
x
(ctg x)¢ = - 1
sin2
x
(ln x)¢ = 1
õ
(1 / õ)¢ = - 1
õ2
(Öõ)¢ = 1
2 Öõ
(õ)¢ = 1
Ëîãàðèôìû.
logàâ =
ñ
logà 1
= 0
logà à
= 1
logà (m n) = logà m + logà n
logà m = logà m - logà n
n
logà m n = n logà m
logà n Öm =
1 logà m
n
logàâ =
logñâ
logñ à
Îñíîâíûå òðèãîíîìåòðè÷åñêèå òîæäåñòâà
sin2x + cos2x = 1
tg x = sin x
cos x
ctg x = cos x
sin x
1 + ctg2 x = 1
sin2
x
1 + tg2 x = 1
cos2
x
tg
x ctg x = 1
Ôîðìóëû ñëîæåíèÿ è âû÷èòàíèÿ
sin (a
± b) = sina cosb
± cosa sinb
cos (a
± b) = cosa cosb
± sina sinb
tg (a
± b) = (tga ± tgb)
(1 + tga
tgb)
ctg (a
± b) = ctga ctgb
+ 1
ctgb
± ctga
sina + sinb = 2 sin (a + b) cos (a
- b)
2
2
sina
- sinb = 2 cos (a + b) sin (a
- b)
2
2
cosa
+ cosb = 2 cos (a + b) cos (a - b)
2
2
cosa - cosb = -
2 sin (a + b) sin (a - b)
2
2
tga
± tgb = sin (a
± b)
cosa
cosb
ctga
± ctgb = sin (b
± a)
sina sinb
sin2a
- sin2b = cos2b
- cos2a =
sin (a + b) sin (a
- b)
cos2a
- sin2b = cos2b
- sin2a =
cos (a + b) cos (a
- b)
Ñâÿçü ìåæäó òðèãîíîìåòðè÷åñêèìè ôóíêöèÿìè
sina
= ± Ö1 - cos2a
sina
= tga
±
Ö1 + tg2a
sina
= 1
±
Ö1 + ctg2a
cosa
= ± Ö1 - sin2a
cosa
= 1
±
Ö1 + tg2a
cosa
= ctga
±
Ö1 + ctg2a
tga
= sina
±
Ö1 - sin2a
tga
= ± Ö1 - cos2a
cosa
tga
= 1
ctga
ctga = ± Ö1 - sin2a
sina
ctga
= cosa
±
Ö1 - cos2a
ctga = 1
tga
Ôîðìóëû ïðåîáðàçîâàíèÿ ïðîèçâåäåíèÿ
sina
sinb = cos (a - b)
- cos (a + b)
2
cosa
cosb = cos (a - b)
+ cos (a + b)
2
sina
cosb = sin (a + b) + sin (a
- b)
2
tga
tgb = tga + tgb
ctga
+ ctgb
ctga
tgb = ctga + tgb
tga
+ ctgb
ctga
ctgb = ctga + ctgb
tga + tgb
Ôîðìóëû äâîéíûõ óãëîâ
sin2a
= 2 sina cosa
sina
= 2 sin (a) cos (a)
cos2a
= cos2a - sin2a =
= 1 -
2sin2a =
= 2cos2a
- 1
tg2a
= 2 tga
1 -
tg2a
= 2
ctga
- tga
tga
= 2 tg (a/2)
1 -
tg2 (a/2)
ctg2a
= ctg2a - 1
2 ctga
= ctga
- tga
2
ctga
= ctg2 (a/2) - 1
2 ctg (a/2)
sin x = a
x = (-1)n arksin a
+ pn
cos x = a
x = ±
arkcos a + 2pn
tg x = a
x = arktg a + pn
ctg x = a
x
= arkctg a + pn
Ôîðìóëû ïðèâåäåíèÿ
sin (p /2 - a) = + cosa
sin (p /2 + a) = + cosa
sin (p - a) = + sina
sin (p + a) = - sina
sin (3p/2
- a) = -
cosa
sin (3p
/2 + a) = - cosa
sin (2p - a) = - sina
sin (2p + a) = + sina
----------------
cos (p/2
- a) = + sina
cos (p/2
+ a) = - sina
cos (p - a) = - cosa
cos (p
+ a) = - cosa
cos (3p/2
- a) = - sina
cos (3p/2
+ a) = + sina
cos (2p - a) = + cosa
cos (2p
+ a) = + cosa
-----------------
tg (p/2
- a) = + ctga
tg (p/2
+ a) = - ctga
tg (p - a) = - tga
tg (p
+ a) = + tga
tg (3p/2
- a) = + ctga
tg (3p/2
+ a) = - ctga
tg (2p - a) = - tga
tg (2p
+ a) = + tga
-------------
ctg (p/2
- a) = + tga
ctg (p/2
+ a) = - tga
ctg (p
- a) = - ctga
ctg (p
+ a) = + ctga
ctg (3p/2
- a) = + tga
ctg (p/2
+ a) = - tga
ctg (2p
- a) = - ctga
ctg
(2p + a) = + ctga
sin (-
a) = - sina
cos (-
a) = cosa
tg (-
a) = - tga
 ïðÿìîóãîëüíîì òðåóãîëüíèêå
a2 + b2 =
c2
a = c sina
a = b tga
b = c cosa
òåîðåìà ñèíóñîâ:
a = b = c
sina
sinb sing
òåîðåìà êîñèíóñîâ:
a2 = b2 +
c2 - 2 bc cosa
S = ½ ab
Ïëîùàäè ôèãóð
Ïðÿìîóãîëüíèê
S = a b = ½ d1 d2
sina,
d1 è d2 - äèàãîíàëè
a - óãîë ïåðåñå÷åíèÿ äèàãîíàëåé
Ïàðàëëåëîãðàìì
S = a h = a b sina
S = ½ d1 d2
sina
Òðàïåöèÿ
S = a + b h = ½ d1
d2 sina
2
Êðóã
S = l r = p
r2
2
ÒÐÅÓÃÎËÜÍÈÊ
S = ½ ah =
½ ab sina
Ôîðìóëà Ãåðîíà:
S = Ö p (p - a) (p - b) (p -
c)
p = a +b + c
2
Ïëîùàäü
òðåóãîëüíèêà îïèñàííîãî îêðóæíîñòüþ:
S = a b c
4r
Ïëîùàäü
òðåóãîëüíèêà ñ âïèñàííîé îêðóæíîñòüþ:
S = ½ r P
ãäå Ð – ïåðèìåòð
ðàäèóñ îïèñàííîé
îêðóæíîñòè:
R = a b c
4S
ðàäèóñ âïèñàííîé
îêðóæíîñòè:
r = 2S
a + b + c
äëèíà îêðóæíîñòè:
l = 2pr
Êâàäðàò
S = a2 = d2/2
Ðîìá
S = a2 sina
= ah = ½ dD
ãäå d - ìàëàÿ
äèàãîíàëü
D - áîëüøàÿ äèàãîíàëü
Îáúåìû òåë:
Ïàðàëëåëåïèïåä
V = Sîñí h
Êóá
V = abc = a3
Ïðèçìà
V = Sîñí h = S^ñå÷ l
l - ãðàíü ïðèçìû
Ïèðàìèäà
V = 1/3 Sîñí h
Öèëèíäð
V = Sîñí h = p r2 h = 1/4p
d2 h
r - ðàäèóñ îñíîâàíèÿ
d - äèàìåòð îñíîâàíèÿ
Êîíóñ
V = 1/3 Sîñí h = 1/3 p r2 h
Øàð
V = 4/3 p r3
Ïëîùàäè ïîâåðõíîñòåé
Ïðèçìà
Sï = Sáîê
+ 2Sîñí
Sáîê
= ph = S^ñå÷
l
p =
a + b +c
Êóá
Sï = 6a2
Ïèðàìèäà ÷åòûðåõóãîëüíàÿ
Sï = Sáîê
+ Sîñí
Sáîê = ½ Pîñí h
h – âûñîòà áîêîâîé ãðàíè
Ïèðàìèäà òðåóãîëüíàÿ
Sï = Sáîê
+ Sîñí
Sáîê = Sîñí cosj
j - óãîë
íàêëîíà ãðàíè
Öèëèíäð
Sï = Sáîê + Sîñí
Sáîê
= 2p rh
Sîñí
= 2pr (h + r)
Êîíóñ
Sï = Sáîê
+ Sîñí
Sáîê = prl
Sîñí = pr (l + r)
Ïàðàëëåëåïèïåä
Sï = Sáîê + 2Sîñí
Sáîê
= Pîñí l
Øàð
S = 4 pr2
Çíà÷åíèÿ óãëîâ
a 0 p/6
p/4 p/3 p/2
p
sin
0 ½ Ö2/2 Ö3/2 1 0
cos
1 Ö3/2 Ö2/2 ½
0 -1
tg
0 1/Ö3 1 Ö3
- 0
ctg
- Ö3 1 1/Ö3
0 -
