Question

Oblicz wartość pozostałych funkcji trygonometrycznych kąta ostrego alfa:
zad.1 sin alfa 12/13
zad.2 cos alfa 2/5
zad.3 tg alfa 8/15

Answer 1

Będziemy korzystać z podanych wzorów:
sin²α + cos²α = 1
tgα = sinα/cosα

ZADANIE 1

sinα = 12/13
(12/13)² + cos²α = 1
cos²α = 1 - 144/169 = 25/169 ⇒ cosα = 5/13
tgα = (12/13)/(5/13) = 12/5
ctgα = 5/12

ZADANIE 2

cosα = 2/5
sin²α + (2/5)² = 1
sin²α = 1 - 4/25 = 21/25 ⇒ sinα = √21/5
tgα = (√21/5)/(2/5) = √21/2
ctgα = 2/√21 = (2√21)/21

ZADANIE 3

tgα = 8/15
tgα = sinα/cosα
8/15 = sinα/cosα
sinα = 8cosα/15
(8cosα/15)² + cos²α = 1
64cos²α/225 + cos²α = 1 | · 225
64cos²α + 225cos²α = 225
289cos²α = 225
cos²α = 225/289 ⇒ cosα = 15/17
sinα = 8cosα/15 = (8 · 15/17)/15 = (120/17)/15 = 8/17
ctgα = 15/8