Question

Решить с развёрнутым ответом

Answer 1

1.558
а) Используем формулу понижения степени:
${cos}^{2}x = frac{1 + cos2x}{2} \ 1 + cos2x = 2 {cos}^{2} x \ \ 1 + cos120 = 2 {cos}^{2}60 = 2 times {( frac{1}{2}) }^{2} = 2 times frac{1}{4} = frac{1}{2}$
б)
${sin}^{2} x = frac{1 - cos2x}{2} \ 1 - cos2x = 2 {sin}^{2} x \ \ 1 - cos120 = 2 {sin}^{2} 60 = 2 {( frac{ sqrt{3} }{2} )}^{2} = 2 times frac{3}{4} = frac{3}{2}$
1.559.
а) Используем формулу косинуса двойного угла:
$cos {}^{2} a - {sin}^{2} a = cos2a$
${cos}^{2} 67.5 - {sin}^{2} 67.5 = cos(2 times 67.5) = cos135 = cos(180 - 45) = - cos45 = - frac{ sqrt{2} }{2}$
б)
${sin}^{2} 75 - {cos}^{2} 75 = - ( {cos}^{2} 75 - {sin}^{2} 75) = - cos(2 times 75) = - cos150 = - cos(180 - 30) = - ( - cos30) = cos30 = frac{ sqrt{3} }{2}$
1.560
а)
${( {cos}^{2} frac{pi}{8} - {sin}^{2} frac{pi}{8}) }^{2} = {(cos(2 times frac{pi}{8})) }^{2} = {(cos frac{pi}{4}) }^{2} = {( frac{ sqrt{2} }{2} )}^{2} = frac{2}{4} = frac{1}{2}$
б) Используем формулу синуса двойного угла
$sin2 alpha = 2sin alpha cos alpha$
${(2sin frac{pi}{12}cos frac{pi}{12}) }^{ - 1} = frac{1}{2sin frac{pi}{12} cos frac{pi}{12} } = frac{1}{sin(2 times frac{pi}{12}) } = frac{1}{sinfrac{pi}{6} } = frac{1}{ frac{1}{2} } = 2$