Đáp án:
\(\begin{array}{l}
2,\\
\left[ \begin{array}{l}
\cos x = \dfrac{{\sqrt {21} }}{5};\,\,\tan x = \dfrac{{2\sqrt {21} }}{{21}};\,\,\,\,\cot x = \dfrac{{\sqrt {21} }}{2}\\
\cos x = - \dfrac{{\sqrt {21} }}{5};\,\,\tan x = - \dfrac{{2\sqrt {21} }}{{21}};\,\,\,\,\cot x = - \dfrac{{\sqrt {21} }}{2}
\end{array} \right.
\end{array}\)
\(\begin{array}{l}
3,\\
A = 4
\end{array}\)
Giải thích các bước giải:
Bài 2:
Ta có:
\(\begin{array}{l}
{\sin ^2}x + {\cos ^2}x = 1\\
\Leftrightarrow {\left( {\dfrac{2}{5}} \right)^2} + {\cos ^2}x = 1\\
\Leftrightarrow \dfrac{4}{{25}} + {\cos ^2}x = 1\\
\Leftrightarrow {\cos ^2}x = \dfrac{{21}}{{25}}\\
\Rightarrow \left[ \begin{array}{l}
\cos x = \dfrac{{\sqrt {21} }}{5}\\
\cos x = - \dfrac{{\sqrt {21} }}{5}
\end{array} \right.\\
TH1:\,\,\,\,\,\cos x = \dfrac{{\sqrt {21} }}{5}\\
\Rightarrow \left\{ \begin{array}{l}
\tan x = \dfrac{{\sin x}}{{\cos x}} = \dfrac{2}{{\sqrt {21} }} = \dfrac{{2\sqrt {21} }}{{21}}\\
\cot x = \dfrac{{\cos x}}{{\sin x}} = \dfrac{{\sqrt {21} }}{2}
\end{array} \right.\\
TH2:\,\,\,\,\,\,\cos x = - \dfrac{{\sqrt {21} }}{5}\\
\Rightarrow \left\{ \begin{array}{l}
\tan x = \dfrac{{\sin x}}{{\cos x}} = - \dfrac{2}{{\sqrt {21} }} = - \dfrac{{2\sqrt {21} }}{{21}}\\
\cot x = \dfrac{{\cos x}}{{\sin x}} = - \dfrac{{\sqrt {21} }}{2}
\end{array} \right.
\end{array}\)
Vậy \(\left[ \begin{array}{l}
\cos x = \dfrac{{\sqrt {21} }}{5};\,\,\tan x = \dfrac{{2\sqrt {21} }}{{21}};\,\,\,\,\cot x = \dfrac{{\sqrt {21} }}{2}\\
\cos x = - \dfrac{{\sqrt {21} }}{5};\,\,\tan x = - \dfrac{{2\sqrt {21} }}{{21}};\,\,\,\,\cot x = - \dfrac{{\sqrt {21} }}{2}
\end{array} \right.\)
Bài 3:
\(\begin{array}{l}
\sin x = \cos \left( {90^\circ - x} \right)\\
{\sin ^2}x + {\cos ^2}x = 1\\
A = {\sin ^2}10^\circ + {\sin ^2}20^\circ + {\sin ^2}30^\circ + {\sin ^2}40^\circ + {\sin ^2}50^\circ + {\sin ^2}60^\circ + {\sin ^2}70^\circ + {\sin ^2}80^\circ \\
= {\sin ^2}10^\circ + {\sin ^2}20^\circ + {\sin ^2}30^\circ + {\sin ^2}40^\circ + {\cos ^2}\left( {90^\circ - 50^\circ } \right) + {\cos ^2}\left( {90^\circ - 60^\circ } \right) + {\cos ^2}\left( {90^\circ - 70^\circ } \right) + {\cos ^2}\left( {90^\circ - 80^\circ } \right)\\
= {\sin ^2}10^\circ + {\sin ^2}20^\circ + {\sin ^2}30^\circ + {\sin ^2}40^\circ + {\cos ^2}40^\circ + {\cos ^2}30^\circ + {\cos ^2}20^\circ + {\cos ^2}10^\circ \\
= \left( {{{\sin }^2}10^\circ + {{\cos }^2}10^\circ } \right) + \left( {{{\sin }^2}20^\circ + {{\cos }^2}20^\circ } \right) + \left( {{{\sin }^2}30^\circ + {{\cos }^2}30^\circ } \right) + \left( {{{\sin }^2}40^\circ + {{\cos }^2}40^\circ } \right)\\
= 1 + 1 + 1 + 1\\
= 4
\end{array}\)
