1) (1 - cosα + cos2α)/(sinα - sin2α) = (1 - cosα + 2cos²α - 1)/(sinα - 2sinαcosα) = ( 2cos²α - cosα)/(sinα(1 - 2cosα)) = - cosα(1 - 2cosα)/(sinα(1 - 2cosα)) = - cosα/sinα = -ctgα.
2) sin10°sin30°sin50°sin70° = (2sin10°cos10°sin30°sin50°sin70°)/(2cos10°) = (sin20°sin30°sin50°sin70°)/(2cos10°) = (2sin20°sin30°sin50°cos20°)/(4cos10°) = (sin40°sin30°sin50°)/(4cos10°) = (2sin40°sin30°cos40°)/(8cos10°) = (sin80°sin30°)/(8cos10°) = (cos10°sin30°)/(8cos10°) =(sin30°)/8 = 0,5/8 = 1/16.
3) sinπ/16cos³π/16 - sin³π/16cosπ/16 = sinπ/16cosπ/16(cos²π/16 - sin²π/16) = 0,5·2sinπ/16cosπ/16(cos2π/16) = 0,5sin2π/16cosπ/8 = 0,5sinπ/8cosπ/8 = 0,25·2sinπ/8cosπ/8 = 0,25sin2π/8 = 0,25sinπ/4 = 0,25·√2/2 = √2/8
4) sin(2α - π)/(1 - sin(3π/2 + 2α)) = -sin(2α)/(1 + cos(2α)) = (-2sinαcosα)/(2cos²α) = (-sinα)/(cosα) = -tgα.
5) (2cos²α - sin2α)/(2sin²α - sin2α) = (2cos²α - 2sinαcosα)/(2sin²α - 2sinαcosα) = -(2cosα(sinα - cosα))/(2sinα(sinα - cosα)) = -(cosα)/(sinα) = -ctgα = 4
6) sin36°/sin12° - cos36°/cos12° = (sin36°·cos12° - sin12°·cos36°)/(sin12°·cos12°) = (2sin(36° - 12°))/(2sin12°·cos12°) = (2sin24°)/(sin24°) = 2
7) cos92°·cos2° + 0,5sin4° + 1 = 0,5(cos(92° - 2°) + cos(92° + 2°)) + 0,5sin4° + 1 = 0,5(cos(90°) + cos(94°)) + 0,5sin4° + 1 = 0,5cos94° + 0,5sin4° + 1 = 0,5cos94° + 0,5sin4° + 1 = 0,5(cos94° + sin4°) + 1 = 0,5(-sin4° + sin4°) + 1 = 0 + 1 = 1.
1 кг помидор стоит у грн
х грн 100% а = 150*х/100 = 1.5х (грн) - цена 1 кг огурцов псле подорожания.
а грн 150%
у грн 100% в 80*х/100 = 0,8х (грн) - цена 1 кг помидор после
в грн 80% после уценки.
По условию задачи составим систему.
4х + 3у = 17
2 * 1,5х + 5 * 0,8у = 18, 3х + 4у = 18
Первое уравнение умножим на 4 и отнимем второе уравнение умноженное на 3.
Получим 16х + 12у - 9х - 12у = 68 - 54 ===> 7x = 14, ===> x = 2 (грн)
4 * 2 + 3у = 17 ===> 3y = 17 - 8 3y = 9 ===> y = 3 (грн)
ответ. 1 кг огурцов стоит 2 (грн), 1 кг помидор стоит 3 (грн).
1) (1 - cosα + cos2α)/(sinα - sin2α) = (1 - cosα + 2cos²α - 1)/(sinα - 2sinαcosα) = ( 2cos²α - cosα)/(sinα(1 - 2cosα)) = - cosα(1 - 2cosα)/(sinα(1 - 2cosα)) = - cosα/sinα = -ctgα.
2) sin10°sin30°sin50°sin70° = (2sin10°cos10°sin30°sin50°sin70°)/(2cos10°) = (sin20°sin30°sin50°sin70°)/(2cos10°) = (2sin20°sin30°sin50°cos20°)/(4cos10°) = (sin40°sin30°sin50°)/(4cos10°) = (2sin40°sin30°cos40°)/(8cos10°) = (sin80°sin30°)/(8cos10°) = (cos10°sin30°)/(8cos10°) =(sin30°)/8 = 0,5/8 = 1/16.
3) sinπ/16cos³π/16 - sin³π/16cosπ/16 = sinπ/16cosπ/16(cos²π/16 - sin²π/16) = 0,5·2sinπ/16cosπ/16(cos2π/16) = 0,5sin2π/16cosπ/8 = 0,5sinπ/8cosπ/8 = 0,25·2sinπ/8cosπ/8 = 0,25sin2π/8 = 0,25sinπ/4 = 0,25·√2/2 = √2/8
4) sin(2α - π)/(1 - sin(3π/2 + 2α)) = -sin(2α)/(1 + cos(2α)) = (-2sinαcosα)/(2cos²α) = (-sinα)/(cosα) = -tgα.
5) (2cos²α - sin2α)/(2sin²α - sin2α) = (2cos²α - 2sinαcosα)/(2sin²α - 2sinαcosα) = -(2cosα(sinα - cosα))/(2sinα(sinα - cosα)) = -(cosα)/(sinα) = -ctgα = 4
6) sin36°/sin12° - cos36°/cos12° = (sin36°·cos12° - sin12°·cos36°)/(sin12°·cos12°) = (2sin(36° - 12°))/(2sin12°·cos12°) = (2sin24°)/(sin24°) = 2
7) cos92°·cos2° + 0,5sin4° + 1 = 0,5(cos(92° - 2°) + cos(92° + 2°)) + 0,5sin4° + 1 = 0,5(cos(90°) + cos(94°)) + 0,5sin4° + 1 = 0,5cos94° + 0,5sin4° + 1 = 0,5cos94° + 0,5sin4° + 1 = 0,5(cos94° + sin4°) + 1 = 0,5(-sin4° + sin4°) + 1 = 0 + 1 = 1.