wzór |
dane |
szukane |
`A + B + C = 180°` |
suma kątów w trójkącie |
`A, B` |
`C = 180° - A - B` |
`a/(sin A) = b/(sin B) = c/(sin C) = 2R` |
wzór sinusów, twierdzenie Snelliusa |
`a, A, B` |
`b = (a sin B)/(sin A)` |
`a, b, A` |
`sin B = (b sin A)/a` |
`c^2 = a^2 + b^2 - 2ab cos C` |
wzór cosinusów, twierdzenie Carnota, twierdzenie al-Kasziego, uogólnione twierdzenie Pitagorasa |
`a, b, C` |
`c = sqrt(a^2 + b^2 - 2ab cos C)` |
`a, b, c` |
`cos A = (b^2 + c^2 - a^2)/(2bc)` |
`cos B = (a^2 + c^2 - b^2)/(2ac)` |
`cos C = (a^2 + b^2 - c^2)/(2ab)` |
`c = a cos B + b cos A` |
wzór rzutów |
`a, b, A, B` |
`c = a cos B + b cos A` |
`a, b, A, C` |
`c = (b - a cos C)/(cos A)` |
`a, b, c, A` |
`cos B = (c - b cos A)/a` |
`cos C = (b - c cos A)/a` |
`(a + b)/c = (cos (A - B)/2)/(cos (A + B)/2)` |
wzory Mollweide’a |
`a, b, A, B` |
`c = (a + b) (cos (A + B)/2)/(cos (A - B)/2)` |
`(a - b)/c = (sin (A - B)/2)/(sin (A + B)/2)` |
`c = (a - b) (sin (A + B)/2)/(sin (A - B)/2)` |
`(a + b)/(a - b) = ("tg" (A + B)/2)/("tg" (A - B)/2)` |
wzór tangensów Nepera / Regiomontana |
`a, b, C` |
1 |
`A + B = 180° - C` |
2 |
`"tg" (A - B)/2 = (a - b)/(a + b) "tg" (A+ B)/2` |
`"tg" (A - B)/2 = (a - b)/(a + b) "ctg" C/2` |
3 |
`A = ((A + B) + (A - B))/2` |
4 |
`B = ((A + B) - (A - B))/2` |
`"tg" C = (c sin B)/(a - c cos B) = (c sin A)/(b - c cos A)` |
wzór tangensów |
`a, A, B` |
`c = a cos B + (a sin B)/("tg" A)` |
`a, c, B` |
`"tg" A = (a sin B)/(c - a cos B)` |
`"tg" C = (c sin B)/(a - c cos B)` |
`("ctg" A/2)/(p - a) = ("ctg" B/2)/(p - b) = ("ctg" C/2)/(p - c) = 1/r` |
wzór cotangensów |
`a, b, c, A` |
`"ctg" B/2 = (p - b)/(p - a) "ctg" A/2` |
`"ctg" C/2 = (p - c)/(p - a) "ctg" A/2` |
`sin^2 C/2 = ((p - a)(p - b))/(ab)` |
wzory połówkowe |
`a, b, c` |
`sin A/2 = sqrt(((p - b)(p - c))/(bc))` |
`cos A/2 = sqrt((p(p - a))/(bc))` |
`"tg" A/2 = sqrt(((p - b)(p - c))/(p(p - a)))` |
`cos^2 C/2 = (p(p - c))/(ab)` |
`sin B/2 = sqrt(((p - a)(p - c))/(ac))` |
`cos B/2 = sqrt((p(p - b))/(ac))` |
`"tg" B/2 = sqrt(((p - a)(p - c))/(p(p - b)))` |
`"tg"^2 C/2 = ((p - a)(p - b))/(p(p - c))` |
`sin C/2 = sqrt(((p - a)(p - b))/(ab))` |
`cos C/2 = sqrt((p(p - c))/(ab))` |
`"tg" C/2 = sqrt(((p - a)(p - b))/(p(p - c)))` |
`sin A = (2S)/(bc)` |
|
`a, b, c` |
`sin A = 2/(bc) sqrt(p(p - a)(p - b)(p - c))` |
`sin B = 2/(ac) sqrt(p(p - a)(p - b)(p - c))` |
`sin C = 2/(ab) sqrt(p(p - a)(p - b)(p - c))` |
|
|
|
|
wzór |
dane |
szukane |
wysokość trójkąta |
`b, C` |
`h_a = b sin C` |
`c, B` |
`h_a = c sin B` |
`a, C` |
`h_b = a sin C` |
`c, A` |
`h_b = c sin A` |
`a, B` |
`h_c = a sin B` |
`b, A` |
`h_c = b sin A` |
środkowe trójkąta |
`a, b, c` |
`m_a = 1/2 sqrt(2b^2 + 2c^2 - a^2)` |
`m_b = 1/2 sqrt(2a^2 + 2c^2 - b^2)` |
`m_c = 1/2 sqrt(2a^2 + 2b^2 - c^2)` |
dwusieczne trójkąta |
`a, b, c` |
`l_A = (sqrt(bcp(p - a)))/(b + c)` |
`l_B = (sqrt(bcp(p - b)))/(a + c)` |
`l_C = (sqrt(bcp(p - c)))/(a + b)` |
promień okręgu wpisanego |
`a, b, c` |
`r = sqrt(((p - a)(p - b)(p - c))/p)` |
`a, b, c, A` |
`r = (p - a) "tg" A/2` |
`a, b, c, B` |
`r = (p - b) "tg" B/2` |
`a, b, c, C` |
`r = (p - c) "tg" C/2` |
`a, b, c, S` |
`r = S/p` |
`a, b, c, R` |
`r = (abc)/(4pR)` |
promień okręgu opisanego |
`a, A` |
`R = a/(2 sin A)` |
`a, b, c, r` |
`R = (abc)/(4pr)` |
`a, b, c, S` |
`R = (abc)/(4S)` |
pole trójkąta |
`a, h_a` |
`S = 1/2 ah_a` |
`a, b, C` |
`S = 1/2 ab sin C` |
`a, B, C` |
`S = (a^2 sin B sin C)/(2 sin (B + C))` |
`a, b, c` |
`S = sqrt(p(p - a)(p - b)(p - c))` |
`a, b, c, r` |
`S = pr` |
`a, b, c, R` |
`S = (abc)/(4R)` |
`A, B, C, R` |
`S = 2R^2 sin A sin B sin C` |