Odpowiedź:
zad 64
sin²α + cos²α = (a/c)² + (b/c)² = a²/c² + b²/c² = (a² + b²)/c² = (a² + b²)/(a² + b²) = 1
zad 64
a)
(a/c)/(b/c) = a/b
sinα/cosα = tgα
b)
1/(a/b) = b/a
1/(tgα) = ctgα
c)
a/b * b/a = 1
tgα * ctgα = 1
zad 66
b)
cosα = 0,1
sin²α + (0,1)² = 1
sin²α = 1 - (0,1)² = 1 - 0,01 = 0,99
sinα = √0,99 = √(99/100) = √(9 * 11)/10 = 3√11/10
tgα = sinα/cosα = 3√11/10 : 1/10 = 3√11/10 * 10 = 3√11
ctgα = 1/tgα = 1/3√11 = √11/(3 * 11) = √11/33
c)
cosα = 1/5
sin²α + (1/5)² = 1
sin²α = 1 - (1/5)² = 1 - 1/25 = 24/25
sinα = √(24/25) = √24/5 = √(4 * 6)/5 = 2√6/5
tgα = sinα/cosα = 2√6/5 : 1/5 = 2√6/5 * 5 = 2√6
ctgα = 1/tgα = 1/2√6 = √6/(2 * 6) = √6/12
d)
sinα = 24/25
(24/25)² + cos²α = 1
576/625 + cos²α = 1
cos²α = 1 - 576/625 = 49/625
cosα = √(49/625) = 7/25
tgα = sinα/cosα = 24/25 : 7/25 = 24/25 * 25/7 = 24/7 = 3 3/7
ctgα = 1/tgα = 1 : 24/7 = 1 * 7/24 = 7/24
