GoniometrickΘ funkcie
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Vz╗ahy medzi funkciami Funkcie s·Φtu a rozdielu S·Φty a rozdiely funkciφ S·Φiny funkciφ Funkcie nßsobkov uhlov |
Zßkladn² vz╗ah:
Vz╗ahy medzi funkciami
i) eiae-ia = (cosa + isina)(cosa - isina) = cos2a + sin2a = 1ii) tg2a + 1 = (cos2a + sin2a)/cos2a = 1/cos2a
| Funkcia | sina | cosa | tga | cotga |
| sina= | sina |   (1-cos2a)
| tga/(1+tg2a)
| 1/(1+cotg2a)
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| cosa= | (1-sin2a)
| cosa | 1/(1+tg2a)
| cotga/(1+cotg2a)
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| tga= | sina/(1-sin2a)
| (1-cos2a)/cosa
| tga | 1/cotga |
| cotga= | (1-sin2a)/sina
| cosa/(1-cos2a)
| 1/tga | cotga |
Funkcie s·Φtu a rozdielu
ei(a+b) = eiaeibcos(a+b) + isin(a+b) = (cosa + isina)(cosb + isinb) =
= cosacosb - sinasinb + i(cosasinb + sinacosb)
| sin(a b) = cosasinb sinacosb
| sin(2a) = 2sinacosa |
| sin2(a/2) = (1 - cosa)/2 | |
cos(a b) = cosacosb  sinasinb
| cos(2a) = cos2a - sin2a |
| cos2(a/2) = (1 + cosa)/2 | |
| tg(a b) = (tga tgb)/(1 tgatgb)
| tg(2a) = 2tga/(1 - tg2a) |
| tg2(a/2) = (1 - cosa)/(1 + cosa) | |
| cotg(a b) = (cotgacotgb 1)/(cotgb cotga)
| cotg(2a) = (cotg2a - 1)/(2cotga) |
| cotg2(a/2) = (1 + cosa)/(1 - cosa) |
S·Φty a rozdiely funkciφ
cos(a + b) + cos(a - b) = cosacosb - sinasinb + cosacosb + sinasinb == 2cosacosb = |substit·cia a = (x + y)/2, substit·cia b = (x - y)/2| =
= 2cos[(x + y)/2]cos[(x - y)/2] = cosx + cosy
| sina + sinb = 2sin[(a + b)/2]cos[(a - b)/2] |
| sina - sinb = 2cos[(a + b)/2]sin[(a - b)/2] |
| cosa + cosb = 2cos[(a + b)/2]cos[(a - b)/2] |
| cosa - cosb = -2sin[(a + b)/2]sin[(a - b)/2] |
| tga tgb = sin(a b)/(cosacosb)
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| cotga cotgb = sin(b a)/(sinasinb)
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| tga cotgb = cos(a b)/(cosasinb)
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S·Φiny funkciφ
cos(a - b) - cos(a + b) = cosacosb + sinasinb - cosacosb + sinasinb == 2sinasinb
| sinasinb = [cos(a - b) - cos(a + b)]/2 |
| cosacosb = [cos(a - b) + cos(a + b)]/2 |
| sinacosb = [sin(a + b) + sin(a - b)]/2 |
| tgatgb = (tga + tgb)/(cotga + cotgb) |
| cotgacotgb = (cotga + cotgb)/(tga + tgb) |
| tgacotgb = (tga + cotgb)/(cotga + tgb) |
Funkcie nßsobkov uhlov
e3ia = (eia)3(cos3a + isin3a) = (cosa + isina)3
cos3a + isin3a = cos3a + 3icos2asina + 3i2cosasin2a + i3sin3a
cos3a + isin3a = cos3a - 3cosasin2a + i(3cos2asina + sin3a)
| sin3a = 3sina - 4sin3a |
| sin4a = 8sinacos3a - 4sinacosa |
| sin5a = 16sinacos4a - 12sinacos2a + sina |
| sinna = nsinacosn-1a - C(3,n)sin3acosn-3a + C(5,n)sin5acosn-5a - .. |
| cos3a = 4cos3a - 3cosa |
| cos4a = 8cos4a - 8cos2a + 1 |
| cos5a = 16cos5a - 20cos3a + 5cosa |
| cosna = cosna - C(2,n)sin2acosn-2a + C(4,n)sin4acosn-4a - .. |
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