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Quan hệ giữa các giá trị lượng giác của hai góc đặc biệt

Quan hệ giữa các giá trị lượng giác của hai góc đặc biệt (bù nhau, phụ nhau, đối nhau, hơn kém (pi ), hơn kém (frac{pi }{2}), …)
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1. Lý thuyết

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

\(\sin ( - \alpha ) =  - \sin \alpha \); \(\tan ( - \alpha ) =  - \tan \alpha \)
\(\cos ( - \alpha ) = \cos \alpha \); \(\cot ( - \alpha ) =  - \cot \alpha \)

+ Hai góc phụ nhau \(\alpha \) và \({90^ \circ } - \alpha \)

\(\sin \left( {{{90}^ \circ } - \alpha } \right) = \cos \alpha \); \(\tan \left( {{{90}^ \circ } - \alpha } \right) = \cot \alpha \)
\(\cos \left( {{{90}^ \circ } - \alpha } \right) = \sin \alpha \); \(\cot \left( {{{90}^ \circ } - \alpha } \right) = \tan \alpha \)

+ Hai góc bù nhau \(\alpha \) và \({180^ \circ } - \alpha \)

\(\sin \left( {{{180}^ \circ } - \alpha } \right) = \sin \alpha \); \(\tan \left( {{{180}^ \circ } - \alpha } \right) =  - \tan \alpha \)
\(\cos \left( {{{180}^ \circ } - \alpha } \right) =  - \cos \alpha \); \(\cot \left( {{{180}^ \circ } - \alpha } \right) =  - \cot \alpha \)

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

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

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

\(\sin \left( {{{180}^ \circ } + \alpha } \right) =  - \sin \alpha \); \(\tan \left( {{{180}^ \circ } + \alpha } \right) = \tan \alpha \)
\(\cos \left( {{{180}^ \circ } + \alpha } \right) =  - \cos \alpha \); \(\cot \left( {{{180}^ \circ } + \alpha } \right) = \cot \alpha \)

Chú ý: Với \(k \in \mathbb{Z}\), ta có:

\(\sin \left( {2k{{.180}^ \circ } + \alpha } \right) = \sin \alpha \); \(\tan \left( {k{{.180}^ \circ } + \alpha } \right) = \tan \alpha \)
\(\cos \left( {2k{{.180}^ \circ } + \alpha } \right) = \cos \alpha \); \(\cot \left( {k{{.180}^ \circ } + \alpha } \right) = \cot \alpha \)
 

2. Ví dụ minh họa

Ví dụ 1. Cho tam giác ABC, khi đó ta có

\(\sin A = \sin ({180^ \circ } - A) = \sin (B + C)\)\(\sin \frac{A}{2} = \cos \left( {{{90}^ \circ } - \frac{A}{2}} \right) = \cos \left( {\frac{{B + C}}{2}} \right)\)

Ví dụ 2. Tính các giá trị lượng giác \(\𒈔sin {570^ౠ \circ },\cos ( - {1035^ \circ }),\tan ({1500^ \circ }).\)

\(\begin{array}{l}\sin {570^ \circ } = \sin ({360^ \circ } + {180^ \circ } + {30^ \circ }) = \sin ({180^ \circ } + {30^ \circ }) =  - \sin {30^ \circ } =  - \frac{1}{2}\\\cos ( - {1035^ \circ }) = \cos ( - {3.2.180^ \circ } + {45^ \circ }) = \cos ({45^ \circ }) = \frac{{\sqrt 2 }}{2}\\\tan ({1500^ \circ }) = \tan ({8.180^ \circ } + {60^ \circ }) = \tan ({60^ \circ }) = \sqrt 3 .\end{array}\)
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