Bài tập 43 trang 214 SGK Toán 10 NC

a)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
\cos {75^0}\cos {15^0}\\
 = \frac{1}{2}\left( {\cos \left( {{{75}^0} - {{15}^0}} \right) + \cos \left( {{{75}^0} + {{15}^0}} \right)} \right)\\
 = \frac{1}{2}\left( {\cos {{60}^0} + \cos {{90}^0}} \right) = \frac{1}{4}
\end{array}\\
\begin{array}{l}
\sin {75^0}\sin {15^0}\\
 = \frac{1}{2}\left( {\cos \left( {{{75}^0} - {{15}^0}} \right) - \cos \left( {{{75}^0} + {{15}^0}} \right)} \right)\\
 = \frac{1}{2}\left( {\cos {{60}^0} - \cos {{90}^0}} \right) = \frac{1}{4}
\end{array}
\end{array}\) 

Vậy \(\cos {75^0}\cos {15^0} = \sin {75^0}\sin {15^0} = \frac{1}{4}\)

b)

\(\begin{array}{l}
\cos {75^0}\sin {15^0} = \sin {15^0}\cos {75^0}\\
 = \frac{1}{2}\left[ {\sin \left( {{{15}^0} - {{75}^0}} \right) + \sin \left( {{{15}^0} + {{75}^0}} \right)} \right]\\
 = \frac{1}{2}\left[ {\sin \left( { - {{60}^0}} \right) + \sin {{90}^0}} \right]\\
 = \frac{{2 - \sqrt 3 }}{4}
\end{array}\)

c)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
\sin {75^0}\cos {15^0}\\
 = \frac{1}{2}\left[ {\sin \left( {{{75}^0} - {{15}^0}} \right) + \sin \left( {{{75}^0} + {{15}^0}} \right)} \right]
\end{array}\\
{ = \frac{1}{2}\left[ {\sin {{60}^0} + \sin {{90}^0}} \right]}\\
{ = \frac{{2 + \sqrt 3 }}{4}}
\end{array}\)

d)

\(\begin{array}{*{20}{l}}
\begin{array}{l}
\cos \alpha \sin \left( {\beta  - \gamma } \right)\\
 = \frac{1}{2}\left[ {\sin \left( {\alpha  + \beta  - \gamma } \right) - \sin \left( {\alpha  - \beta  + \gamma } \right)} \right]
\end{array}\\
\begin{array}{l}
\cos \beta \sin \left( {\gamma  - \alpha } \right)\\
 = \frac{1}{2}\left[ {\sin \left( {\beta  + \gamma  - \alpha } \right) - \sin \left( {\beta  - \gamma  + \alpha } \right)} \right]
\end{array}\\
\begin{array}{l}
\cos \gamma \sin \left( {\alpha  - \beta } \right)\\
 = \frac{1}{2}\left[ {\sin \left( {\gamma  + \alpha  - \beta } \right) - \sin \left( {\gamma  - \alpha  + \beta } \right)} \right]
\end{array}\\
\begin{array}{l}
 \Rightarrow \cos \alpha \sin \left( {\beta  - \gamma } \right) + \cos \beta \sin \left( {\gamma  - \alpha } \right)\\
 + \cos \gamma \sin \left( {\alpha  - \beta } \right) = 0,\forall \alpha ,\beta ,\gamma 
\end{array}
\end{array}\)

-- Mod Toán 10 HỌC247