Bài tập 39 trang 213 SGK Toán 10 NC

a) Ta có:

\(\begin{array}{l}
\cos {75^0} = \cos \left( {{{45}^0} + {{30}^0}} \right)\\
 = \cos {45^0}.\cos {30^0} - \sin {45^0}.\sin {30^0}\\
 = \frac{{\sqrt 2 }}{4}\left( {\sqrt 3  - 1} \right)
\end{array}\)

\(\begin{array}{l}
\sin {75^0} = \sin \left( {{{45}^0} + {{30}^0}} \right)\\
 = \sin {45^0}.\cos {30^0} + \cos {45^0}.\sin {30^0}\\
 = \frac{{\sqrt 2 }}{2}\left( {\frac{{\sqrt 3 }}{2} + \frac{1}{2}} \right) = \frac{{\sqrt 2 }}{4}\left( {\sqrt 3  + 1} \right)\\
\tan {75^0} = \frac{{\sqrt 3  + 1}}{{\sqrt 3  - 1}} = 2 + \sqrt 3 \\
\cot {75^0} = 2 - \sqrt 3 
\end{array}\)

b) Ta có:

\(\begin{array}{*{20}{l}}
{\cos {{15}^0} = \cos \left( {{{45}^0} - {{30}^0}} \right)}\\
\begin{array}{l}
 = \cos {45^0}.\cos {30^0} + \sin {45^0}.\sin {30^0}\\
 = \frac{{\sqrt 2 }}{4}\left( {\sqrt 3  + 1} \right)
\end{array}
\end{array}\)

\(\begin{array}{l}
\sin {15^0} = \sin \left( {{{45}^0} - {{30}^0}} \right)\\
 = \sin {45^0}.\cos {30^0} - \cos {45^0}.\sin {30^0}\\
 = \frac{{\sqrt 2 }}{2}\left( {\frac{{\sqrt 3 }}{2} - \frac{1}{2}} \right) = \frac{{\sqrt 2 }}{4}\left( {\sqrt 3  - 1} \right)\\
\tan {15^0} = \frac{{\sqrt 3  - 1}}{{\sqrt 3  + 1}} = 2 - \sqrt 3 \\
\cot {15^0} = 2 + \sqrt 3 
\end{array}\)

-- Mod Toán 10 HỌC247