a) sin (90⁰ - α ) = 3/10 < => cos α = 3/10 sin²α = 1 - cos²α = 1 - (3/10)² = 1 - 9/100 = 91/100 sin α = √91 : √100 = √91/10 tg α = sin α : cos α = √91/10 : (3/10) = √91/10 *(10/3) = √91/3 ctg α = 1/ tg α = 3/√91 -------------------------------- b) cos ( 90⁰ - α) = √3/2 < => sin α = √3/2 cos²α = 1 - sin²α = 1 - (√3/2)² = 1 - 3/4 = 1/4 cos α = 1/2 tg α = (√3/2) : (1/2) = √3 ctg α = 1 /√3 α = 60⁰ -------------------------------------- c) ctg ( 90⁰ - α) = 2/5 <=> tg α = 2/5 tg = y/x --> y = 2 ; x = 5 r² = x² + y² = 5² + 2² =25 + 4 = 29 r = √29 sin α = y/r = 2/√29 cos α = x/r = 5/√29 ctg α = 1/ (2/5) = 5/2 = 2,5 ===================================
Zad. 1) a) sin(90°- α) = 3/10 sin(90°- α) = cosα cosα = 0,3 sin²α + cos²α = 1 sin²α + 0,3² = 1 sin²α = 1 - 0,3² sin²α = 1 - 0,09 sin²α = 0,91 sin²α - 0,91 = 0 (sinα - √0,91)(sinα + √0,91) = 0 sinα - √91/10 = 0 lub sinα + √91/10 = 0 sinα = √91/10 lub sinα = - √91/10 tgα = sinα/cosα tgα = (√91/10)/0,3 lub tgα = - (√91/10)/0,3 tgα = √91/3 lub tgα = - √91/3 ctgα = 1/tgα ctgα = 3√91/91 lub ctgα = - 3√91/91 b) cos(90°- α) = √3/2 cos(90°- α) = sinα sinα = √3/2 sin²α + cos²α = 1 (3/4)² + cos²α = 1 cos²α = 1 - (3/4)² cos²α = 1 - 9/16 cos²α = 7/16 cos²α - (√7/4)² = 0 (cosα - √7/4)(cosα + √7/4) = 0 cosα - √7/4 = 0 lub cosα + √7/4 = 0 cosα = √7/4 lub cosα = - √7/4 tgα = sinα/cosα tgα = (√3/2)/(√7/4) lub tgα = - (√3/2)/(√7/4) tgα = 2√21/7 lub tgα = - 2√21/7 ctgα = 1/tgα ctgα = √21/6 lub ctgα = - √21/6 c) ctg(90°- α) = 2/5 ctg(90°- α) = tgα tgα = 0,4 ctgα = 2,5 tgα = sinα/cosα sinα = 0,4cosα sin²α + cos²α = 1 (0,4cosα)² + cos²α = 1 1,16cos²α = 1 /:(116/100) cos²α = 100/116 cosα = 5√29/29 lub cosα = - 5√29/29 sinα = 2√29/29 lub sinα = - 2√29/29
