1. Упростите:

\(\frac{{\sin \left( {\frac{\pi }{2} - x} \right) \cdot \cos (\pi - x)}}{{{{\sin }^2}x - 1}}.\)

Решение:

\(\begin{array}{l}\sin \left( {\frac{\pi }{2} - x} \right) = \cos x;\\cos(\pi - x) = - \cos x;\\{\sin ^2}x - 1 = - (1 - {\sin ^2}x) = - {\cos ^2}x;\\\frac{{\sin \left( {\frac{\pi }{2} - x} \right) \cdot \cos (\pi - x)}}{{{{\sin }^2}x - 1}} = \frac{{ - {{\cos }^2}x}}{{ - {{\cos }^2}x}} = 1.\end{array}\)

2. Вычислите:

\(\cos {14^0}\cos {16^0} - \sin {14^0}\sin {16^0}.\)

Решение:

Применим формулу \(\cos (\alpha + \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta ;\) тогда:

\(\cos {14^0}\cos {16^0} - \sin {14^0}\sin {16^0} = \cos {30^0} = \frac{{\sqrt 3 }}{2}.\)

3. Вычислите:

\(k = \frac{{\cos {{64}^0}\cos {4^0} - \cos {{86}^0}\cos {{26}^0}}}{{\cos {{71}^0}\cos {{41}^0} - \cos {{49}^0}\cos {{19}^0}}}.\)

Решение:

По формулам приведения получаем:

\(\begin{array}{l}\cos {64^0} = \cos ({90^0} - {26^0}) = \sin {26^0};\\\cos {86^0} = \cos ({90^0} - {4^0}) = \sin {4^0};\\\cos {71^0} = \cos ({90^0} - {19^0}) = \sin {19^0};\\\cos {49^0} = \cos ({90^0} - {41^0}) = \sin {41^0}.\end{array}\)

Тогда:

\(\begin{array}{l}k = \frac{{\cos {{64}^0}\cos {4^0} - \cos {{86}^0}\cos {{26}^0}}}{{\cos {{71}^0}\cos {{41}^0} - \cos {{49}^0}\cos {{19}^0}}} = \\ = \frac{{\sin {{26}^0}\cos {4^0} - \sin {4^0}\cos {{26}^0}}}{{\sin {{19}^0}\cos {{41}^0} - \sin {{41}^0}\cos {{19}^0}}} = \frac{{\sin {{22}^0}}}{{ - \sin {{22}^0}}} = - 1.\end{array}\)

4. Вычислите:

\(10\sin \left( {2arctg\frac{1}{2}} \right).\)

Решение:

Обозначим \(arctg\frac{1}{2} = \alpha ,\) тогда \(tg\alpha = \frac{1}{2},\)

\(10\sin (2\alpha ) = 10\frac{{2tg\alpha }}{{1 + t{g^2}\alpha }} = 10\frac{{2 \cdot \frac{1}{2}}}{{1 + \frac{1}{4}}} = 8.\)

5. Найти значение выражения:

\(\frac{{5\cos \left( {\frac{{3\pi }}{5} + \alpha } \right)}}{{4\sin (2\pi - \alpha )}},\) если \(\alpha = \frac{{7\pi }}{6}.\)

Решение:

\(\frac{{5\cos \left( {\frac{{3\pi }}{5} + \alpha } \right)}}{{4\sin (2\pi - \alpha )}} = \frac{{5\sin \alpha }}{{ - 4\sin \alpha }} = - 1.25.\)