Bài tự luận · Bài 14
Dạng 1. Tính giá trị của các biểu thức chứa giá trị lượng giác
Bài tự luận Công thức lượng giác · Bài 14
(CTST11) Không sử dụng máy tính cầm tay, tính giá trị của các biểu thức sau:
a) $\sin6^{\circ}\cos12^{\circ}\cos24^{\circ}\cos48^{\circ}$;
b) $\cos68^{\circ}\cos78^{\circ}+\cos22^{\circ}\cos12^{\circ}+\cos190^{\circ}$.
a) $\sin6^{\circ}\cos12^{\circ}\cos24^{\circ}\cos48^{\circ}$;
b) $\cos68^{\circ}\cos78^{\circ}+\cos22^{\circ}\cos12^{\circ}+\cos190^{\circ}$.
Xem lời giải
Lời giải
a) Đặt $A=\sin6^{\circ}\cos12^{\circ}\cos24^{\circ}\cos48^{\circ}$. Ta có:
$\begin{array}{l} \cos6^{\circ}\cdot A=\cos6^{\circ}\sin6^{\circ}\cos12^{\circ}\cos24^{\circ}\cos48^{\circ}=\frac{1}{2}\sin12^{\circ}\cos12^{\circ}\cos24^{\circ}\cos48^{\circ} \\ =\frac{1}{4}\sin24^{\circ}\cos24^{\circ}\cos48^{\circ}=\frac{1}{8}\sin48^{\circ}\cos48^{\circ}=\frac{1}{16}\sin96^{\circ}=\frac{1}{16}\cos6^{\circ} \end{array}$
Suy ra $A=\frac{1}{16}$.
b) $\begin{array}{l} \cos68^{\circ}\cos78^{\circ}+\cos22^{\circ}\cos12^{\circ}+\cos190^{\circ}=\sin22^{\circ}\sin12^{\circ}+\cos22^{\circ}\cos12^{\circ}-\cos10^{\circ} \\ =\cos\left( 22^{\circ}-12^{\circ} \right)-\cos10^{\circ}=0 \end{array}$
$\begin{array}{l} \cos6^{\circ}\cdot A=\cos6^{\circ}\sin6^{\circ}\cos12^{\circ}\cos24^{\circ}\cos48^{\circ}=\frac{1}{2}\sin12^{\circ}\cos12^{\circ}\cos24^{\circ}\cos48^{\circ} \\ =\frac{1}{4}\sin24^{\circ}\cos24^{\circ}\cos48^{\circ}=\frac{1}{8}\sin48^{\circ}\cos48^{\circ}=\frac{1}{16}\sin96^{\circ}=\frac{1}{16}\cos6^{\circ} \end{array}$
Suy ra $A=\frac{1}{16}$.
b) $\begin{array}{l} \cos68^{\circ}\cos78^{\circ}+\cos22^{\circ}\cos12^{\circ}+\cos190^{\circ}=\sin22^{\circ}\sin12^{\circ}+\cos22^{\circ}\cos12^{\circ}-\cos10^{\circ} \\ =\cos\left( 22^{\circ}-12^{\circ} \right)-\cos10^{\circ}=0 \end{array}$