Заполни пропуски в решении

Если точка \(A (x;y)\) лежит на окружности радиуса \(R\) с центром в начале координат и угол \(\alpha (\angle AOB)\) отложен от положительной полуоси \(Ox\) в верхнюю полуплоскость, то

\(\sin \alpha = \dfrac{y}{R}\) ,

\(\cos \alpha = \dfrac{x}{R}\) ,

\(\tg \alpha = \dfrac{y}{x}\) ,

\(\ctg \alpha = \dfrac{x}{y}\) .

\(\sin 0 \degree = 0\) , \(\cos 0 \degree = 1\) , \(\tg 0 \degree = 0\) ;

\(\sin 90 \degree = 1\) , \(\cos 90 \degree = 0\) , \(\ctg 90 \degree = 0\) ;

\(\sin 180 \degree = 0\) , \(\cos 180 \degree = - 1\) , \(\tg 180 \degree = 0\) .

Для любого угла \(\alpha\) , \(0 \degree \lt \alpha \lt 180 \degree\) , справедливы равенства

\(\sin (180 \degree - \alpha ) = \sin \alpha\) , \(\cos (180 \degree - \alpha ) = - \cos \alpha\) , \(\ctg (180 \degree - \alpha ) = - \ctg \alpha\) .

\(x\_{A\_1} = -x\_A\) ; \(\dfrac{x\_{A\_1}}{R} = \dfrac{x\_A}{R}\)

\(\cos(180 \degree - \alpha) = - \cos \alpha\)

\(\dfrac{y\_{A\_1}}{R} = \dfrac{y\_A}{R}\)

\(\sin(180 \degree - \alpha) = \sin \alpha\)

\(x\_{A\_1} = -x\_A\) , \(y\_{A\_1} = y\_A\) ;

\(\dfrac{x\_{A\_1}}{y\_{A\_1}} = - \dfrac{x\_A}{y\_A}\)

\(\ctg (180 \degree - \alpha) = - \ctg \alpha\)

Для любого угла \(\alpha\) , \(0 \degree \lt \alpha \lt 90 \degree\) , \(90 \degree \lt \alpha \lt 180 \degree\) , справедливо равенство

\(\tg (180 \degree - \alpha) = - \tg \alpha.\)

Найди значение:

1. \(\sin 150 \degree\) ;
2. \(\cos 120 \degree\) ;
3. \(\tg 135 \degree\) ;
4. \(\ctg 120 \degree\) ;
5. \(\sin 135 \degree\) ;
6. \(\cos 150 \degree\) ;
7. \(\tg 120 \degree\) ;
8. \(\ctg 135 \degree\) .

Решение.

1. \(\sin 150 \degree = \sin (180 \degree - 30 \degree ) = \sin 30 \degree =\) [ ];
2. \(\cos 120 \degree = \cos (180 \degree -\) [ ] \(\degree ) = - \cos\) [ ] \(\degree = - \) [ ];
3. \(\tg 135 \degree = \tg (\) [ ] \(\degree - 45 \degree ) =- \tg 45 \degree =\) [ ];
4. \(\ctg 120 \degree = \ctg (\) [ ] \(\degree - 60 \degree ) =- \ctg\) [ ] \(\degree =\) [ ];
5. \(\sin 135 \degree = \sin (\) [ ] \(\degree -\) [ ] \(\degree ) =\) [ ] \(=\) [ ];
6. \(\cos 150 \degree = \cos (\) [ ] \(\degree -\) [ ] \(\degree ) =\) [ ] \(=\) [ ];
7. \(\tg 120 \degree = \tg (\) [ ] \(\degree -\) [ ] \(\degree ) =\) [ ] \(=\) [ ];
8. \(\ctg 135 \degree = \ctg (\) [ ] \(\degree - \) [ ] \(\degree ) =\) [ ] \(=\) [ ].