Короткий довiдник з математики. Оформив: Виспянський Iгор ( Версiя 1.00, Веб-сайт:
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1 Короткий довiдник з математики Оформив: Виспянський Iгор ( Версiя 1.00, Веб-сайт:
2 Змiст 1 Властивостi степеня Многочлени 3 3 Властивостi арифметичних коренiв 3 4 Тотожнi перетворення тригонометричних виразiв Формули додавання Формули подвiйного аргументу Формули половинного аргументу Формули перетворення суми в добуток Формули перетворення добутку в суму Спiввiдношення мiж sin x, cos x i tg x Формули зведення Прогресiї Арифметична прогресiя Геометрична прогресiя Комбiнаторика i бiном Ньютона 6 7 Логарифми. Показниковi i логарифмiчнi рiвняння 7 8 Тригонометричнi рiвняння 8 9 Трикутник Довiльний трикутник Прямокутний трикутник Рiвностороннiй трикутник Чотирикутник Паралелограм Ромб Прямокутник Квадрат Трапецiя Многокутник Описаний многокутник Правильний многокутник Коло, круг Додатковi спiввiдношення мiж елементами фiгур 11 1
3 14 Многогранники Довiльна призма Пряма призма Прямокутний паралелепiпед Довiльна пiрамiда Куб Правильна пiрамiда Довiльна зрiзана пiрамiда Правильна зрiзана пiрамiда Тiла обертання Цилiндр Конус Куля, сфера Кульовий сегмент Кульовий сектор Зрiзаний конус Деякi спiввiдношення мiж елементами фiгур Формули диференцiювання Первiснi Прямокутна декартова система координат Прямокутна декартова система координат на площинi Прямокутна декартова система координат у просторi Властивостi степеня a 0 = 1 a x a y = a x+y a x a y = ax y (a x ) y = a x y (a b) x = a x b x ( a b ) x = ax b x a x = 1 a x
4 Многочлени де x 1 i x коренi рiвняння a b = (a b)(a + b) (a + b) = a + ab + b (a b) = a ab + b (a + b) 3 = a 3 + 3a b + 3ab + b 3 (a + b) 3 = a 3 + b 3 + 3ab(a + b) (a b) 3 = a 3 3a b + 3ab b 3 (a b) 3 = a 3 b 3 + 3ab(a b) a 3 + b 3 = (a + b)(a ab + b ) a 3 b 3 = (a b)(a + ab + b ) ax + bx + c = a(x x 1 )(x x ), ax + bx + c = 0 3 Властивостi арифметичних коренiв n ab = n a n b n a n a b = n, (b 0) b ( n a) k = n a k n k a = kn a n a = nk a k ( n a) n = a, (a 0) n a < n b, якщо 0 a < b a = a = { a, при a 0, a, при a < 0 n a n = a n+1 a = n+1 a, a 0 3
5 4 Тотожнi перетворення тригонометричних виразiв 4.1 Формули додавання sin x + cos x = 1 tg x = sin x cos x, x π (n + 1), n Z ctg x = cos x sin x, x πk, k Z tg x ctg x = 1, x πn, n Z 1 + tg x = 1 cos x, x π (n + 1) 1 + ctg x = 1 sin x, x πn) sin(x + y) = sin x cos y + cos x sin y sin(x y) = sin x cos y cos x sin y cos(x + y) = cos x cos y sin x sin y cos(x y) = cos x cos y + sin x sin y tg(x + y) = tg(x y) = tg x + tg y 1 tg x tg y tg x tg y 1 + tg x tg y, x, y, x y π + πn, n Z 4. Формули подвiйного аргументу sin x = sin x cos x cos x = cos x sin x = cos x 1 = 1 sin x tg x = tg x 1 tg x x π 4 + π k, k Z, x π + πn, n Z 4.3 Формули половинного аргументу tg x = sin x = 1 cos x cos x = 1 + cos x sin x 1 + cos x = 1 cos x sin x, x π + πn, n Z 4
6 4.4 Формули перетворення суми в добуток sin x + sin y = sin x + y sin x sin y = cos x + y cos x + cos y = cos x + y cos x cos y = sin x + y tg x tg y = tg x + tg y = sin(x + y) cos x cos y cos x y sin x y cos x y sin x y sin(x y) cos x cos y, x, y π + πn, n Z 4.5 Формули перетворення добутку в суму sin x sin y = 1 (cos(x y) cos(x + y)) cos x cos y = 1 (cos(x y) cos(x + y)) sin x cos y = 1 (sin(x y) sin(x + y)) 4.6 Спiввiдношення мiж sin x, cos x i tg x 4.7 Формули зведення sin x = tg x 1 + tg x, x (n + 1)π cos x = 1 tg x 1 + tg x, x (n + 1)π π u + α π + α 3π + α α π α π α 3π α sin u cos α sin α cos α sin α cos α sin α cos α cos u sin α cos α sin α cos α sin α cos α sin α tg u ctg α tg α ctg α tg α ctg α tg α ctg α ctg u tg α ctg α tg α ctg α tg α ctg α tg α 5
7 5 Прогресiї 5.1 Арифметична прогресiя S n = a 1 + a n a n = a 1 + d(n 1) n = a 1 + d(n 1) n a k = a k 1 + a k+1, k =, 3,..., n 1 a k + a m = a p + a q, де k + m = p + q a 1 - перший член; d - рiзниця; n - число членiв; a n - n-й член; S n - сума n перших членiв. 5. Геометрична прогресiя b n = b 1 q n 1 b 1 - перший член, q - знаменник (q 0), n - число членiв, b n - n-й член, S n - сума n перших членiв. S n = b 1(1 q n ), q 1 1 q b k = b k 1 b k+1, k =, 3,..., (n 1) b k b m = b p b q, де k + m = p + q S = b 1, де q < 1 1 q S - сума нескiнченної геометричної прогресiї. 6 Комбiнаторика i бiном Ньютона P n = 1... (n 1) n = n! Cn m n! = m!(n m)!, C0 n = 1 Cn m = Cn n m = Cn m + Cn m+1 = Cn+1 m+1 A m n = P n Cn m n! = (n m)! (a + b) n = C 0 na n + C 1 na n 1 b C k na n k b k C n nb n (a + b) n = a n + na n 1 n(n 1)... (n k + 1) b a n k b k b n k! Cna k n k b k = T k+1 - (k + 1)-й член у розкладi бiнома (k = 0, 1,,..., n) C 0 n + C 1 n C n n = n 6
8 7 Логарифми. Показниковi i логарифмiчнi рiвняння y = a x, a > 0, a 1 y = log a x x > 0, x = a log a x log a a = 1 log a 1 = 0 log a (bc) = log a b + log a c log a b c = log a b log a c log a x p = p log a x log a x = log b x log b a log a b = 1 log b a log b b log b a = 1 log a b = log a p b p = p log a p b a log a b = b log c a log a b = log c b log a α b β = β α log a b a log c b = b log c a log a α b = log a b log a a α = 1 α log a b log c a log c b = log c b log c a log c b log c a = log c a log c b a f(x) = b g(x) f(x) log c a = g(x) log c b 7
9 8 Тригонометричнi рiвняння 9 Трикутник 9.1 Довiльний трикутник sin x = 0, x = πn, n Z sin x = 1, x = π + πn sin x = 1, x = π + πn cos x = 0, x = π + πn cos x = 1, x = πn cos x = 1, x = π + πn tg x = 0, x = πn ctg x = 0, x = π + πn sin x = a, x = ( 1) n arcsin a + πn cos x = a, x = ± arccos a + πn tg x = a, x = arctg a + πn ctg x = a, x = arcctg a + πn S = S = 1 ah a S = 1 bc sin α p(p a)(p b)(p c) r = S p R = abc 4S a = b + c bc cos α (теорема косинусiв) a sin α = b sin β = c sin γ = R (теорема синусiв) 8
10 9. Прямокутний трикутник S = 1 ab S = 1 ch c r = a + b c R = c a + b = c (теорема Пiфагора) a c h c = h c a c a = a c b c b = b c a c, b c - проекцiї катетiв на гiпотенузу. a = c sin α = c cos β = b tg α = b ctg β 9.3 Рiвностороннiй трикутник 10 Чотирикутник b c S = a 3 4 r = a 3 6 R = a 3 3 S = 1 d 1d sin ϕ 10.1 Паралелограм S = ah a = ab sin α = 1 d 1d sin ϕ 10. Ромб S = ah a = a sin α = 1 d 1d 9
11 10.3 Прямокутник S = ab = 1 d 1d sin ϕ 10.4 Квадрат S = a = d 10.5 Трапецiя S = a + b h = lh, де l = a + b 11 Многокутник 11.1 Описаний многокутник S = pr 11. Правильний многокутник 1 Коло, круг a 3 = R 3 a 4 = R a 6 = R S = na nr c = πr S = πr Сектор (l- довжина дуги, яка обмежує сектор, α- радiанна мiра центрального кута, n - градусна мiра центрального кута). l = πrn 180 = rα S = πr n 360 = 1 r α 10
12 13 Додатковi спiввiдношення мiж елементами фiгур три медiани трикутника перетинаються в однiй точцi, яка дiлить кожну медiану у вiдношенннi : 1, починаючи вiд вершини трикутника медiана трикутника обчислюється за формулою m a = 1 (b + c ) a сторона трикутника обчислюється за формулою a = 3 де m a, m b, m c медiани (m b + m c) m a, бiсектриса дiлить сторону трикутника на вiдрiзки, пропорцiйнi двом iншим його сторонам бiсектриса трикутника обчислюється за формулою l c = ab a 1 b 1 бiсектриса трикутника визначається через його сторони a, b i c за формулою ab(a + b + c)(a + b c) l c = a + b для всякого трикутника залежнiсть мiж висотами h a, h b, h c i радiусом r вписаного кола визначається за формулою 1 h a + 1 h b + 1 h c = 1 r площа S рiвнобiчної трапецiї, дiагоналi якої взаємно перпендикулярнi, дорiвнює квадрату її висоти, тобто S = h висота рiвнобiчної трапецiї, в яку можна вписати коло, є середнiм геометричним її основ h = ab a + b = c (c бiчне ребро) 11
13 14 Многогранники 14.1 Довiльна призма S бiч = Pпер l Pпер периметр перпендикулярного перерiзу. V = S H 14. Пряма призма S бiч = P l 14.3 Прямокутний паралелепiпед S бiч = P H V = a b c 14.4 Довiльна пiрамiда V = 1 3 S H 14.5 Куб V = a 3 d = a Правильна пiрамiда S бiч = 1 P l V = 1 3 S H 14.7 Довiльна зрiзана пiрамiда V = 1 3 h (S 1 + S + S 1 S ) 14.8 Правильна зрiзана пiрамiда S бiч = 1 (P 1 + P ) l 1
14 15 Тiла обертання 15.1 Цилiндр S бiч = π R H V = π R H = S H 15. Конус S бiч = π R l V = 1 3 π R H = 1 3 S H 15.3 Куля, сфера S = 4π R V = 4 3 π R Кульовий сегмент S = π R h V = π h (R 1 3 h) 15.5 Кульовий сектор V = 3 π R h 15.6 Зрiзаний конус V = 1 3 π h (R 1 + R 1 R + R) S = π(r 1 + R )l 16 Деякi спiввiдношення мiж елементами фiгур площу паралелограма можна обчислити за такими формулами D C A B S = AC BD 4 tg A 13
15 S = AB AD tg AOD справджуються такi спiввiдношення мiж елементами сфери i вписаного в неї конуса l = R sin α l = RH α - кут мiж твiрною i площиною основи, l - довжина твiрної конуса. 17 Формули диференцiювання c = 0 (x) = 1 (x α ) = αx α 1 (sin x) = cos x (cos x) = sin x (tg x) = 1 cos x (ctg x) = 1 sin x (e x ) = e x (a x ) = a x ln a (ln x) = 1 x (log a x) = 1 x ln a ( x) = 1 x (f 1 (x) ± f (x)) = f 1(x) ± f (x) (f 1 (x) f (x)) = f 1(x) f (x) + f 1 (x) f (x) (c f(x)) = c f (x) (f(kx + b)) = k f (kx + b) ( ) f1 (x) = f 1 (x)f (x) f 1 (x)f (x) f (x) f (x) (f(u(x))) = f u(u) u (x) 14
16 Формула Ньютона-Лейбнiца має вигляд b a Площа криволiнiйної трапецiї S = f(x)dx = F (b) F (a) b a f(x)dx Рiвняння дотичної до графiка ф-цiї y = f(x) має вигляд y y 0 = f (x 0 )(x x 0 ) 18 Первiснi x α dx = xα+1 α C sin xdx = cos x + C cos xdx = sin x + C 1 cos dx = tg x + C x 1 sin dx = ctg x + C x 1 dx = ln x + C x e x dx = e x + C a x dx = ax ln a + C f(ax + b)dx = 1 f(ax + b) + C a 15
17 19 Прямокутна декартова система координат 19.1 Прямокутна декартова система координат на площинi Вiдстань мiж точками A 1 (x 1 ; y 1 ) i A (x ; y ) визначається за ф-лою A 1 A = Координати середини вiдрiзка (x x 1 ) + (y y 1 ) x = x 1 + x ; y = y 1 + y Рiвняння прямої з кутовим коефiцiєнтом i початковою ординатою має вигляд y = kx + q k = tg α, q значення ординати точки перетину прямої з вiссю O y. Загальне рiвняння прямої ax + by + c = 0 Рiвняння прямих, паралельних вiдповiдно осям O y i O x, мають вигляд x = a; y = b Умови паралельностi i перпендикулярностi прямих y 1 = kx 1 + q 1 i y = kx + q вiдповiдно мають вигляд k 1 = k ; k 1 k = 1. (x x 0 ) + (y y 0 ) = R рiвняння кiл з радiусом R i з центром вiдповiдно в точках O(0; 0) i C(x 0 ; y 0 ). Для y = ax + bx + c xв = b a 19. Прямокутна декартова система координат у просторi ā = a 1 + a + a 3 (a 1 ; a ; a 3 ) + (b 1 ; b ; b 3 ) = (a 1 + b 1 ; a + b ; a 3 + b 3 ) ā 0 = ā ā одиничний вектор. λ(a 1 ; a ; a 3 ) = (λa 1 ; λa ; λa 3 ) āb = ā b cos ϕ 16
18 Скалярний добуток векторiв ā(a 1 ; a ; a 3 ) i b(b 1 ; b ; b 3 ) cos ϕ = Умова перпендикулярностi або āb = a 1 b 1 + a b + a 3 b 3. ā = āā = ā, ā = a a 1 b 1 + a b + a 3 b 3 a 1 + a + a 3 āb = 0 a 1 b 1 + a b + a 3 b 3 = 0 b 1 + b + b 3 Загальне рiвняння площини, перпендикулярної до вектора n(a; b; c) має вигляд ax + by + cz + d = 0 Рiвняння площини, перпендикулярної до вектора n(a; b; c), яка проходить через точку (x 0 ; y 0 ; z 0 ) має вигляд a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 17
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