формулы по математике

sinx
cosx
tg x
ctg x
0
0
1
0
1 cos 2 x
2
5
3
300°
-3/2
1/2
-3
-1/3
11
6
330°
-1/2
3/2
-1/3
-3
7
4
315°
-2/2
2/2
-1
-1
2π
360°
0
1
0
6. sin 2x = 2sin x*cos x
11. cos2x =
2. tg x*ctg x = 1
7. cos 2x = cos2x – sin2x
12. sin2x =
1 cos 2 x
2
23. cos x + cos y = 2 cos
8. cos 2x = 2cos2 - 1
13.cos3x =
3 cos x cos 3 x
4
24. cos x - cos y =  2 sin
9. cos 2x = 1 – sin2x
14. sin3x =
3 sin x sin 3 x
4
25. tgx  tgy = cos x cos y
1
3. 1 + tg2x = cos 2 x
1
4. 1 + ctg2x = sin 2 x
1
5. tg x + ctg x = sin xcos x
2 tgx
15. sin(x ± y) = sin x*cos y ± cos x *siny
16. cos(x + y) = cos x*cosy - sinx*siny
19. sinx* siny =
20. cosx*cosy =
21. sinx*cosy =
1
sin x  y   sin x  y 
2
Теорема Виета: x2 + px + q = 0;
b 2  4ac
X1,2 =
x1 + x2 = - p;
 b  b 2  4ac
2a
x1 . x2 = q.

4
0
\
7
4
3
2
x
1  cos x
26. sin  
2
2
x
1  cos x

2
2
x
1  cos x
28. tg  
2
1  cos x
27. cos
x y
x y
sin
2
2
x
2
1  tg 2
1
cosx  y   cosx  y  1. sin x = 0,
2
Квадратное уравнение: ax2 + bx + c = 0 ax2 + bx + c = a(x – x1)(x – x2)
D=
2tg
2. sin x = 1,
tgx  tgy
18. tg(x ± y) = 1 tgxtgy
/
sin(x  y )
1
cosx  y   cosx  y  31. sin x 
2
17. cos(x - y) = cos x*cosy + sinx*siny
2
/
5
4
x y
x y
cos
2
2
29. sin 3x = 3 sin x – 4 sin3x
10. tg 2x = 1 tg 2 x

 косинус
x y
x y
cos
2
2
1. sin2x + cos2x = 1
22.sinx ± siny = 2 sin
3
4\
синус
0
Тригонометрия
π
2
3
5
7
5
4
3




3
4
6
6
4
3
2
6
4
3
2
30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270°
1/2 2/2 3/2 1
0
-1/2 -2/2 -3/2 -1
3/2 2/2 1/2
-1/2 -2/2 -3/2 -1
3/2 2/2 1/2 0
-3/2 -2/2 -1/2 0
1
-1
0
1/3
3
-3
-1/3
1/3 1
3
1
1
3
1/3 0
-1/3 -1
-3
3
1/3 0
1. tg x = 0,
cos 3x = 4 cos3x – 3 cos x
x
2
32. cos x 
2 x
1  tg
2
1  tg 2
x
2
x = πn.
x=

+ 2 πn.
2

+ 2 πn.
2
4. sin x = a, x = (-1)narcsin a + πn
3. sin x = -1,
30.
x=-
x = πn.
3. tg x = a,

+ πn.
2
1. cos x = 0,
x=
2. cos x = 1,
x = 2 πn.
3. cos x = -1,
x = π + 2 πn.
4. cos x = a, x = ± arcos a + 2 πn

2. tg x = ± 1,
x=±
+ πn.
4
x = arctg a + πn.
Правила дифференцирования
(u + v)/ = u / + v /
Правила интегрирования
(ku)/ = ku/
u
u v  uv 
( ) 
v
v2
(uv)/ = u /v + uv /
(f(kx + m))/ = kf /(kx + m)
1. C = 0
3. (ex) = ex
2. (kx + b)= k
4. (xn) = nxn - 1
1
1
9. (ctg x) = - sin 2 x
1
1
14. (arcsin x) =
11. (logax) = x ln a
1
13. (arctg x) =
1  x2
n
x n 1
1
15. (arccos x) = -
1 x
2
1
2
ab  n a  n b
3.
nm
5.
n
a
km
a

b
 n a k
n
n
a
b
3.

4.

x n 1
n 1
2.
4.
6.
 a
k
n
dx
= ln | x |
x
ax
x
a dx =
ln a
a a
n k
a  nk a
2
12.  x dx 
2 x3
3
1. alogab = b
2. loga a = 1, a 1, a > 0
3. loga 1 = 0
4. logabc = loga b + logac
5. loga
b
= logab – loga c
c
logc b
6. loga bp = p logab
7. logab = logc a
1
9. log 1 b   log a b
8. loga b = logb a
a
dx
6.  sin 2 x = -ctgx
13.  cosx dx = sinx
10. logaxp = ploga | x | p – чет.
7.  ex dx = ex
14.  sin x dx = -cosx
11. logab = x  ax = b
Свойства степени.
2. a1 = a
4. ar : as = ar – s
5. (ar)s = ars
ar  a 
 
br  b 
r
ФСУ
3. ar  as = ar + s
1. a0 = 1 (a0)
Арифметическая
2
2
dx
x
 arcsin
2
a x
a
dx
1
x
 arctg
10.  2
2
x a
a
a
dx
 ln x  x 2  a 2
11.  2
2
x a
6. ar  br = (ab)r
9. a q  a p
q
11. если r > s, то
ar > as при a > 1
ar < as при 0 < a < 1
прогрессии
1. a2 – b2 = (a – b)(a + b)
2. (a + b)2 = a2 + 2ab + b2
p
1
8. a-r = a r
ar > br при r > 0
ar < br при r < 0
 n ak
dx
1
xa
 a ln
2
x a
2
xa
9. 
10. если 0 < a < b, то
Свойства корней
n

xndx =
7.
1 x
1
16. (arcctg x) = 1  x2
17. Уравнение касательной:
y = f(a) + f /(a)(x – a)
1.
2.
8. 
1
10.(ln x) = x
12. ( x n ) = 
dx = x
dx
7. (cos x) = -sin x
8. (tg x) = cos 2 x

5.  cos 2 x = tgx
5. (ax) = ax ln a
6. (sin x) = cos x
1.
Логарифмы
3. (a - b)2 = a2 - 2ab + b2
4. a3 – b3 = (a – b)( a2 + ab + b2)
Геометрическая
5. a3 + b3 = (a + b)( a2 - ab + b2)
6. (a + b)3 = a3 + 3a2 b + 3ab2 + b3
7. (a - b)3 = a3 - 3a2 b + 3ab2 - b3
Симметрические системы
1. bn = b1qn - 1
1. х + у = u, xy = v
1. an = a1 + d(n – 1).
a1  a n = 2a1  d n  1
2. Sn =
2
2
b
b1 1  q n
2. Sn =
= 1 3. x3 + y3 = u3 – 3uv 4. x4 + y4 = u4 – 4u2v + 2v2
1 q
1 q
3. an =
3. bn =
an 1  an 1 
2


bn 1  bn 1
2. x2 + у2 = u2 – 2v
5. x5 + y5 = u5 – 5u3v + 5uv3