Odpowiedź:
zad. 1
a)
cosα = 7/25
cos²α = (7/25)² = 49/625
Korzystamy z zależności :
sin²α + cos²α = 1
sin²α = 1 - cos²α
sin²α = 1 - 49/625 = 625/625 -49/625 = 576/625
sinα = √(576/625) = 24/25
tgα = sinα/cosα = 24/25 : 7/25 = 24/25 * 25/7 = 24/7 = 3 3/7
ctgα = 1/tgα = 7/24
Odp: sinα =24/25 . cosα = 7/25 ,tgα =3 3/7 , ctgα = 7/25
b)
tgα =2√2
sinα/cosα = 2√2
sin²α/cos²α = (2√2)² = 4 * 2 = 8
sin²α = 8cos²α = 8(1 -sin²α) = 8 - 8sin²α
sin²α + 8sin²α =8
9sin²α = 8
sin²α = 8/9
sinα = √(8/9) = √8/3 = 2√2/3
sinα/cosα = 2√2
sinα = 2√2 * cosα
cosα = sinα : 2√2 = 2√2/3 : 2√2 = 1/3
ctgα = 1/tgα = 1/2√2 = √2/(2 * 2) = √2/4
Odp: sinα = 2√2/3 , cosα = 1/3 , tgα = 2√2 , ctgα = √2/4
c)
sinα = 8/17
sin²α = (8/17)² = 64/289
1 - cos²α = 64/289
cos²α = 1 - 64/289 = 289/289 - 64/289 = 225/289
cosα = √(225/289) = 15/17
tgα =sinα/cosα = 8/17 : 15/17 = 8/17 * 17/15 = 8/15
ctgα = 1/tgα = 15/8 = 1 7/8
Odp: sinα = 8/17 , cosα = 15/17 , tgα = 8/15 , ctgα = 1 7/8
zad 2
sinα = 8/17
sin²α = (8/17)² = 64/289
cos²α = 1 - sin²α = 1 - 64/289 = 225/289
cosα = √(225/289) = 15/17
a)
cos²α - sin²α = 225/289 - 64/289 = 161/289
b)
sin²α/(1 + cosα) = 64/289 : (1 + 15/17) = 64/289 : 1 15/17 = 64/289 : 32/17 =
= 64/289 * 17/32 = 2/17 * 1 = 2/17
