Все формулы по тригонометрии
.docxВсе формулы по тригонометрии
Основные тригонометрические тождества
sin2x + cos2x = 1
|
tgx |
= |
sinx |
|
cosx |
|
ctgx |
= |
cosx |
|
sinx |
tgx ctgx = 1
|
tg2x + 1 |
= |
1 |
|
cos2x |
|
ctg2x + 1 |
= |
1 |
|
sin2x |
Формулы двойного аргумента
sin2x = 2sinx cosx
|
sin2x |
= |
2tgx |
= |
2ctgx |
= |
2 |
|
1 + tg2x |
1 + ctg2x |
tgx + ctgx |
cos2x = cos2 - sin2x = 2cos2x - 1 = 1 - 2sin2x
|
cos2x |
= |
1 - tg2x |
= |
ctg2x - 1 |
= |
ctgx - tgx |
|
1 + tg2x |
ctg2x + 1 |
ctgx + tgx |
|
tg2x |
= |
2tgx |
= |
2ctgx |
= |
2 |
|
1 - tg2x |
ctg2x - 1 |
ctgx - tgx |
|
ctg2x |
= |
ctg2x - 1 |
= |
ctgx - tgx |
|
2ctgx |
2 |
Формулы тройного аргумента
sin3x = 3sinx - 4sin3x cos3x = 4cos3x - 3cosx
|
tg3x |
= |
3tgx - tg3x |
|
1 - 3tg2x |
|
ctg3x |
= |
ctg3x - 3ctgx |
|
3ctg2x - 1 |
Формулы половинного аргумента
|
sin2 |
x |
= |
1 - cosx |
|
2 |
2 |
|
cos2 |
x |
= |
1 + cosx |
|
2 |
2 |
|
tg2 |
x |
= |
1 - cosx |
|
2 |
1 + cosx |
|
ctg2 |
x |
= |
1 + cosx |
|
2 |
1 - cosx |
|
tg |
x |
= |
1 - cosx |
= |
sinx |
|
2 |
sinx |
1 + cosx |
|
ctg |
x |
= |
1 + cosx |
= |
sinx |
|
2 |
sinx |
1 - cosx |
Формулы квадратов тригонометрических функций
|
sin2x |
= |
1 - cos2x |
|
2 |
|
cos2x |
= |
1 + cos2x |
|
2 |
|
tg2x |
= |
1 - cos2x |
|
1 + cos2x |
|
ctg2x |
= |
1 + cos2x |
|
1 - cos2x |
|
sin2 |
x |
= |
1 - cosx |
|
2 |
2 |
|
cos2 |
x |
= |
1 + cosx |
|
2 |
2 |
|
tg2 |
x |
= |
1 - cosx |
|
2 |
1 + cosx |
|
ctg2 |
x |
= |
1 + cosx |
|
2 |
1 - cosx |
Формулы кубов тригонометрических функций
|
sin3x |
= |
3sinx - sin3x |
|
4 |
|
cos3x |
= |
3cosx + cos3x |
|
4 |
|
tg3x |
= |
3sinx - sin3x |
|
3cosx + cos3x |
|
ctg3x |
= |
3cosx + cos3x |
|
3sinx - sin3x |
Формулы тригонометрических функций в четвертой степени
|
sin4x |
= |
3 - 4cos2x + cos4x |
|
8 |
|
cos4x |
= |
3 + 4cos2x + cos4x |
|
8 |
Формулы сложения аргументов
sin(α + β) = sinα cosβ + cosα sinβ cos(α + β) = cosα cosβ - sinα sinβ
|
tg(α + β) |
= |
tgα + tgβ |
|
1 - tgα tgβ |
|
ctg(α + β) |
= |
ctgα ctgβ - 1 |
|
ctgα + ctgβ |
sin(α - β) = sinα cosβ - cosα sinβ cos(α - β) = cosα cosβ + sinα sinβ
|
tg(α - β) |
= |
tgα - tgβ |
|
1 + tgα tgβ |
|
ctg(α - β) |
= |
ctgα ctgβ + 1 |
|
ctgα - ctgβ |
Формулы суммы тригонометрических функций
|
sinα + sinβ |
= 2sin |
α + β |
∙ cos |
α - β |
|
2 |
2 |
|
cosα + cosβ |
= 2cos |
α + β |
∙ cos |
α - β |
|
2 |
2 |
(sinα + cosα)2 = 1 + sin2α
|
tgα + tgβ |
= |
sin(α + β) |
|
cosα cosβ |
|
ctgα + ctgβ |
= |
sin(α + β) |
|
sinα sinβ |
Формулы разности тригонометрических функций
|
sinα - sinβ |
= 2sin |
α - β |
∙ cos |
α + β |
|
2 |
2 |
|
cosα - cosβ |
= -2sin |
α + β |
∙ sin |
α - β |
|
2 |
2 |
(sinα - cosα)2 = 1 - sin2α
|
tgα - tgβ |
= |
sin(α - β) |
|
cosα cosβ |
|
ctgα - ctgβ |
= – |
sin(α - β) |
|
sinα sinβ |
Формулы произведения тригонометрических функций
|
sinα ∙ sinβ |
= |
cos(α - β) - cos(α + β) |
|
2 |
|
sinα ∙ cosβ |
= |
sin(α - β) + sin(α + β) |
|
2 |
|
cosα ∙ cosβ |
= |
cos(α - β) + cos(α + β) |
|
2 |
|
tgα ∙ tgβ |
= |
cos(α - β) - cos(α + β) |
= |
tgα + tgβ |
|
cos(α - β) + cos(α + β) |
ctgα + ctgβ |
|
ctgα ∙ ctgβ |
= |
cos(α - β) + cos(α + β) |
= |
ctgα + ctgβ |
|
cos(α - β) - cos(α + β) |
tgα + tgβ |
|
tgα ∙ ctgβ |
= |
sin(α - β) + sin(α + β) |
|
sin(α + β) - sin(α - β) |