Question

Wykaż, że istnieje taki kąt ostry alfa, że cos²α= 5/4 + sin²α

Będe wdzieczna za rozwiazanie! ;)

Answer 1

1.

sin²α+cos²α=1

(1/3)²+cos²α=1

cos²α=1- 1/9

cos²α=8/9

cosα = 2√2 / 3

 

tgα=sinα/cosα

tgα = 1/3 / (2√2 / 3) = 1/3 * 3/2√2 = 1 / 2√2 = √2 / 4 ≠ √2/2

 

2.

 

log(4/3) sinα = -1/2

(4/3)^(-1/2) = sinα

sinα=(3/4)^(1/2)

sinα=√3/2

α =60°

3.

(sinα-cosα)²=sin²α+cos²α-2sinα cosα = 1-2sinα cosα

 

sinα + cosα = √5 / 2                   /()²

sin²α+cos²α +2sinα cosα = 5/4

2sinα cosα = 5/4-1

2sinα cosα = 1/4

-2sinα cosα = -1/4

 

1-2sinα cosα = 1-1/4 = 3/4