Справочный материал по математике. Формулы по тригонометрии. Свойства степен. Арифметическая и геометрическая прогрессии. Основные свойства логарифмов. Площади многоугольников.

Тригонометрия

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1. sin²x + cos²x = 1                    6. sin 2x = 2sin x*cos x          11. cos²x =        15.sinx ± siny =              20.                                      

2. tg x*ctg x = 1                         7. cos 2x = cos²x – sin²x        12. sin²x =        16. cos x + cos y =         21.  

3. 1 + tg²x =                8. cos 2x = 2cos² - 1               13.cos³x =17. cos x - cos y =     22.                                    

4. 1 + ctg²x =            9. cos 2x = 1 – sin²x              14. sin³x =  18. tgx ± tgy =                          

5. tg x + ctg x = 10. tg 2x =                                                                    19.  sin 3x = 3 sin x – 4 sin³x                          23.    cos 3x = 4 cos³x – 3 cos x                   

24. sin(x ±  y) = sin x*cos y ± cos x *siny        25. sinx* siny =   26.          27.

 28. cos(x + y) = cos x*cosy -  sinx*siny           29. cosx*cosy =    1. sin x = 0,         x = πn.                        1. cos x = 0,           x =  + πn.                         

 30. cos(x - y) = cos x*cosy +  sinx*siny                                                                                              2. sin x = 1,         x =  + 2 πn.             2. cos x = 1,           x = 2 πn.                           

 31.  tg(x ± y) =                            32. sinx*cosy =      3. sin x = -1,        x = -  + 2 πn.           3. cos x = -1,         x = π + 2 πn.                        

                                                                             4. sin x = a,   x = (-1)ⁿarcsin a + πn       4. cos x = a,   x = ± arcos a + 2 πn    

Квадратное уравнение:    ax² + bx + c = 0   ax² + bx + c = a(x – x₁)(x – x₂)                               5. tg x = 0,          x = πn.                          2.  tg x = ± 1,      x = ±  + πn.                          

                                                D =       X_1,2 =                                      6.  tg x = a,         x = arctg a + πn.

Теорема Виета:  x² + px + q = 0;       x₁ + x₂ = - p;        x₁ . x₂ = q.

Правила дифференцирования                                     Правила интегрирования                                                               Логарифмы

(u + v)^/ = u ^/ + v ^/           (ku)^/ = ku^/                               1. dx = x                             8.  ò               1. a^log_a^b = b             2. log_a a = 1,  a¹ 1, a > 0

(uv)^/ = u ^/v + uv ^/                              2. xⁿdx =                9. ò                 3. log_a 1 = 0           4. log_abc = log_a b + log_ac

(f(kx + m))^/ = kf ^/(kx + m)                                      3. = ln | x |                       10.  ò                 5. log_a  = log_ab – log_a c  

1.  C¢ = 0      2.  (kx + b)¢= k       3. (e^x)¢ = e^x               4.  a^x dx =                   11.  ò       6. log_a b^p = p log_ab         7. log_ab =

   4. (xⁿ)¢ = nx^{n - 1}             5. (a^x)¢ = a^x ln a                      5.  ò  = tgx               12.  ò                     8. log_a b =         9.

  6.  (sin x)¢ = cos x           7. (cos x)¢ = -sin x                6. ò = -ctgx                 13. ò cosx dx = sinx                         10. log_ax^p = plog_a | x |   p – чет.

  8. (tg x)¢ =     9. (ctg x)¢ = -     7.  ò e^x dx = e^x                           14. ò sin x dx = -cosx                       11. log_ab = x   Û a^x = b

10.(ln x)¢ =             11. (log_ax)¢ =                                      Свойства степени                                         Формулы сокращённого умножения

 12.  () =          13. (arctg x)¢ =         1.  a⁰ = 1  (a¹0)           2. a¹ = a                 3.   a^r × a^s = a^{r + s}             1. a² – b² = (a – b)(a + b) 

14.   (arcsin x)¢ =                                               4. a^r : a^s = a^{r – s}            5.  (a^r)^s = a^rs           6. a^r × b^r = (ab)^r                 2. (a + b)² = a² + 2ab + b²

15.   (arccos x)¢ = -                                            7.                8. a^-r =                  9.        3. (a - b)² = a² - 2ab + b²

16.    (arcctg x)¢ = -                                              10. если  0 < a < b, то                       11. если r > s, то                     4. a³ – b³ = (a – b)( a² + ab + b²)

17.   Уравнение касательной:                                         a^r > b^r  при r > 0                                a^r > a^s при a > 1                    5. a³ + b³ = (a + b)( a² - ab + b²)

       y = f(a) + f ^/(a)(x – a)                                                  a^r < b^r  при r < 0                                a^r < a^s при 0 < a < 1                6. (a + b)³ = a³ + 3a² b + 3ab² + b³

                                                                                                                                                                                                  7. (a - b)³ = a³ - 3a² b + 3ab² - b³

             Свойства корней                                           Арифметическая  и  геометрическая прогрессии                     Симметрические системы

1.          2.                1. a_n = a₁ + d(n – 1).                                      1. b_n = b₁q^{n - 1}                        1. х + у = u,  xy = v    2. x² + у² = u² – 2v

3.           4.                         2. S_n =  =      2. S_n = =    3. x³ + y³ = u³ – 3uv   4. x⁴ + y⁴ = u⁴ – 4u²v + 2v²

5.                 6.                      3. a_n =                                        3. b_n =             5. x⁵ + y⁵ = u⁵ – 5u³v + 5uv³