Задание 12. Найдите значение выражения 1) 6 / (cos^2(74°) + 2 + cos^2(164°)) 2) 10 / (cos^2(92°) + 1 + cos^2(182°)) 3) 26 / (cos^2(59°) + 3 + cos^2(149°)) 4) 20 / (cos^2(33°) + 3 + cos^2(123°))

Краткое пояснение:

Для решения этих примеров будем использовать тригонометрические тождества, в частности, свойства косинуса для смежных углов и основное тригонометрическое тождество \( \sin^2(\alpha) + \cos^2(\alpha) = 1 \).

Пошаговое решение:

1. 1. \[ \frac{6}{\cos^{2}74^{\circ}+2+\cos^{2}164^{\circ}} \]Заметим, что \(\cos164^{\circ} = \cos(180^{\circ}-16^{\circ}) = -\cos16^{\circ}\) и \(\cos74^{\circ} = \sin(90^{\circ}-74^{\circ}) = \sin16^{\circ}\). Следовательно, \(\cos^{2}74^{\circ} = \sin^{2}16^{\circ}\) и \(\cos^{2}164^{\circ} = (- \cos16^{\circ})^2 = \cos^{2}16^{\circ}\).\[ \frac{6}{\sin^{2}16^{\circ}+2+\cos^{2}16^{\circ}} = \frac{6}{(\sin^{2}16^{\circ}+\cos^{2}16^{\circ})+2} = \frac{6}{1+2} = \frac{6}{3} = 2 \]
2. 2. \[ \frac{10}{\cos^{2}92^{\circ}+1+\cos^{2}182^{\circ}} \]Заметим, что \(\cos92^{\circ} = \cos(90^{\circ}+2^{\circ}) = -\sin2^{\circ}\) и \(\cos182^{\circ} = \cos(180^{\circ}+2^{\circ}) = -\cos2^{\circ}\). Следовательно, \(\cos^{2}92^{\circ} = (-\sin2^{\circ})^2 = \sin^{2}2^{\circ}\) и \(\cos^{2}182^{\circ} = (-\cos2^{\circ})^2 = \cos^{2}2^{\circ}\).\[ \frac{10}{\sin^{2}2^{\circ}+1+\cos^{2}2^{\circ}} = \frac{10}{(\sin^{2}2^{\circ}+\cos^{2}2^{\circ})+1} = \frac{10}{1+1} = \frac{10}{2} = 5 \]
3. 3. \[ \frac{26}{\cos^{2}59^{\circ}+3+\cos^{2}149^{\circ}} \]Заметим, что \(\cos59^{\circ} = \sin(90^{\circ}-59^{\circ}) = \sin31^{\circ}\) и \(\cos149^{\circ} = \cos(180^{\circ}-31^{\circ}) = -\cos31^{\circ}\). Следовательно, \(\cos^{2}59^{\circ} = \sin^{2}31^{\circ}\) и \(\cos^{2}149^{\circ} = (-\cos31^{\circ})^2 = \cos^{2}31^{\circ}\).\[ \frac{26}{\sin^{2}31^{\circ}+3+\cos^{2}31^{\circ}} = \frac{26}{(\sin^{2}31^{\circ}+\cos^{2}31^{\circ})+3} = \frac{26}{1+3} = \frac{26}{4} = 6.5 \]
4. 4. \[ \frac{20}{\cos^{2}33^{\circ}+3+\cos^{2}123^{\circ}} \]Заметим, что \(\cos33^{\circ} = \sin(90^{\circ}-33^{\circ}) = \sin57^{\circ}\) и \(\cos123^{\circ} = \cos(180^{\circ}-57^{\circ}) = -\cos57^{\circ}\). Следовательно, \(\cos^{2}33^{\circ} = \sin^{2}57^{\circ}\) и \(\cos^{2}123^{\circ} = (-\cos57^{\circ})^2 = \cos^{2}57^{\circ}\).\[ \frac{20}{\sin^{2}57^{\circ}+3+\cos^{2}57^{\circ}} = \frac{20}{(\sin^{2}57^{\circ}+\cos^{2}57^{\circ})+3} = \frac{20}{1+3} = \frac{20}{4} = 5 \]

Ответ: 1) 2; 2) 5; 3) 6.5; 4) 5