Odpowiedź:
zad 7.1
sinα - cosα = √3/4
Podnosimy obustronnie wyrażenie do kwadratu
(sinα - cosα)² = (√3/4)²
sin²α - 2sinαcosα + cos²α = 3/16
1 - 2sinαcosα = 3/16
- 2sinαcosα = - 1 + 3/16
2sinαcosα = 1 - 3/16 = 16/16 - 3/16 = 13/16
(sinα + cosα)² + 4 = sin²α + 2sinαcosα + cos²α + 4 = 1 + 13/16 + 4 =
= 5 13/16
zad 7.2
sinα + cosα = 5/4
Podnosimy obustronnie wyrażenie do kwadratu
(sinα + cosα)² = (5/4)²
sin²α + 2sinαcosα + cos²α = 25/16
1 + 2sinαcosα = 25/16
2sinαcosα = 25/16 - 1 = 25/16 - 16/16 = 9/16
- 2sinαcosα = - 9/16
(sinα - cosα)² - 3 = sin²α - 2sinαcosα + cos²α - 3 = 1 - 2sinαcosα - 3 =
= 1 - (- 9/16) - 3 = 1 + 9/16 - 3 = 1 9/16 - 2 16/16 = - 1 7/16
