a) Hai góc đối nhau \(\alpha \) và \( - \alpha \)

\(\sin ( - \alpha ) =  - \sin \alpha \);

\(\cos ( - \alpha ) = \cos \alpha \);

\(\tan ( - \alpha ) =  - \tan \alpha \);

\(\cot ( - \alpha ) =  - \cot \alpha \).

b) Hai góc phụ nhau \(\alpha \) và \({90^o} - \alpha \)

\(\sin \left( {{{90}^o} - \alpha } \right) = \cos \alpha \);

\(\cos \left( {{{90}^o} - \alpha } \right) = \sin \alpha \);

\(\tan \left( {{{90}^o} - \alpha } \right) = \cot \alpha \);

\(\cot \left( {{{90}^o} - \alpha } \right) = \tan \alpha \).

c) Hai góc bù nhau \(\alpha \) và \({180^o} - \alpha \)

\(\sin \left( {{{180}^o} - \alpha } \right) = \sin \alpha \);

\(\cos \left( {{{180}^o} - \alpha } \right) =  - \cos \alpha \);

\(\tan \left( {{{180}^o} - \alpha } \right) =  - \tan \alpha \);

\(\cot \left( {{{180}^o} - \alpha } \right) =  - \cot \alpha \).

d) Hai góc \(\alpha \) và \({90^o} + \alpha \)

\(\sin \left( {{{90}^o} + \alpha } \right) = \cos \alpha \);

\(\cos \left( {{{90}^o} + \alpha } \right) =  - \sin \alpha \);

\(\tan \left( {{{90}^o} + \alpha } \right) =  - \cot \alpha \);

\(\cot \left( {{{90}^o} + \alpha } \right) =  - \tan \alpha \).

e) Hai góc \(\alpha \) và \({180^o} + \alpha \)

\(\sin \left( {{{180}^o} + \alpha } \right) =  - \sin \alpha \);

\(\cos \left( {{{180}^o} + \alpha } \right) =  - \cos \alpha \);

\(\tan \left( {{{180}^o} + \alpha } \right) = \tan \alpha \);

\(\cot \left( {{{180}^o} + \alpha } \right) = \cot \alpha \).

1) Cho tam giác ABC, khi đó ta có:

\(\sin A = \sin ({180^o} - A) = \sin (B + C)\);

\(\sin \frac{A}{2} = \cos \left( {{{90}^o} - \frac{A}{2}} \right) = \cos \left( {\frac{{B + C}}{2}} \right)\).

2) Tính các giá trị lượng giác \(\sin {570^o}\), \(\cos ( - {1035^o})\), \(\tan {1500^o}\).

Giải:

\(\sin {570^o} = \sin ({360^o} + {180^o} + {30^o}) = \sin ({180^o} + {30^o}) =  - \sin {30^o} =  - \frac{1}{2}\).

\(\cos ( - {1035^o}) = \cos ( - {3.2.180^o} + {45^o}) = \cos ({45^o}) = \frac{{\sqrt 2 }}{2}\).

\(\tan ({1500^o}) = \tan ({8.180^o} + {60^o}) = \tan ({60^o}) = \sqrt 3 \).