Bài tập nâng cao · Bài 7
Dạng 2. Công thức nhân đôi
Bài tập nâng cao Công thức lượng giác · Bài 7
Tính giá trị của biểu thức sau:
a) $G=\cos\frac{2\pi}{31}.\cos\frac{4\pi}{31}.\cos\frac{8\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}$
b) $H=\sin 5^{\circ}.\sin 15^{\circ}.\sin 25^{\circ}\ldots\sin 75^{\circ}.\sin 85^{\circ}$
c) $I=\cos 10^{\circ}.\cos 20^{\circ}.\cos 30^{\circ}\ldots\cos 70^{\circ}.\cos 80^{\circ}$
d) $K=96\sqrt{3}\sin\frac{\pi}{48}.\cos\frac{\pi}{48}.\cos\frac{\pi}{24}.\cos\frac{\pi}{12}.\cos\frac{\pi}{6}$
e) $L=\cos\frac{\pi}{15}.\cos\frac{2\pi}{15}.\cos\frac{3\pi}{15}.\cos\frac{4\pi}{15}.\cos\frac{5\pi}{15}.\cos\frac{6\pi}{15}.\cos\frac{7\pi}{15}$
f) $M=\sin\frac{\pi}{16}.\cos\frac{\pi}{16}.\cos\frac{\pi}{8}$
a) $G=\cos\frac{2\pi}{31}.\cos\frac{4\pi}{31}.\cos\frac{8\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}$
b) $H=\sin 5^{\circ}.\sin 15^{\circ}.\sin 25^{\circ}\ldots\sin 75^{\circ}.\sin 85^{\circ}$
c) $I=\cos 10^{\circ}.\cos 20^{\circ}.\cos 30^{\circ}\ldots\cos 70^{\circ}.\cos 80^{\circ}$
d) $K=96\sqrt{3}\sin\frac{\pi}{48}.\cos\frac{\pi}{48}.\cos\frac{\pi}{24}.\cos\frac{\pi}{12}.\cos\frac{\pi}{6}$
e) $L=\cos\frac{\pi}{15}.\cos\frac{2\pi}{15}.\cos\frac{3\pi}{15}.\cos\frac{4\pi}{15}.\cos\frac{5\pi}{15}.\cos\frac{6\pi}{15}.\cos\frac{7\pi}{15}$
f) $M=\sin\frac{\pi}{16}.\cos\frac{\pi}{16}.\cos\frac{\pi}{8}$
Xem lời giải
Lời giải
a) Ta có
$G=\cos\frac{2\pi}{31}.\cos\frac{4\pi}{31}.\cos\frac{8\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}$
$\begin{array}{l}\Leftrightarrow G.\sin\frac{2\pi}{31}=\sin\frac{2\pi}{31}.\cos\frac{2\pi}{31}.\cos\frac{4\pi}{31}.\cos\frac{8\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}\\\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{2}\sin\frac{4\pi}{31}.\cos\frac{4\pi}{31}.\cos\frac{8\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}\\\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{4}\sin\frac{8\pi}{31}.\cos\frac{8\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}\\\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{8}\sin\frac{16\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}\\\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{16}\sin\frac{32\pi}{31}.\cos\frac{32\pi}{31}\end{array}$
$\begin{array}{l}\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{32}\sin\frac{64\pi}{31}\\\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{32}\sin\left( \frac{2\pi}{31}+2\pi \right)\\\Leftrightarrow G=\frac{1}{32}\end{array}$
b) Ta có
$\begin{array}{l}H=\sin 5^{\circ}.\sin 15^{\circ}.\sin 25^{\circ}\ldots\sin 75^{\circ}.\sin 85^{\circ}\\=\sin 5^{\circ}.\cos 5^{\circ}.\sin 15^{\circ}.\cos 15^{\circ}.\sin 25^{\circ}.\cos 25^{\circ}.\sin 35^{\circ}.\cos 35^{\circ}.\sin 45^{\circ}\\=\frac{\sqrt{2}}{32}\sin 10^{\circ}.\sin 30^{\circ}.\sin 50^{\circ}.\sin 70^{\circ}\end{array}$
$=\frac{\sqrt{2}}{32.\cos 10^{\circ}}.\cos 10^{\circ}.\sin 10^{\circ}.\sin 30^{\circ}.\sin 50^{\circ}.\cos 20^{\circ}$
$\begin{array}{l}=\frac{\sqrt{2}}{64.\cos 10^{\circ}}.\sin 20^{\circ}.\cos 20^{\circ}.\frac{1}{2}.\sin 50^{\circ}\\=\frac{\sqrt{2}}{256.\cos 10^{\circ}}.\sin 40^{\circ}.\cos 40^{\circ}\\=\frac{\sqrt{2}}{512.\cos 10^{\circ}}.\sin 80^{\circ}=\frac{\sqrt{2}}{512}\end{array}$
c) Ta có
$I=\cos 10^{\circ}.\cos 20^{\circ}.\cos 30^{\circ}\ldots\cos 70^{\circ}.\cos 80^{\circ}$
$\begin{array}{l}=\cos 10^{\circ}.\sin 10^{\circ}.\cos 20^{\circ}.\sin 20^{\circ}.\cos 30^{\circ}.\sin 30^{\circ}\cos 40^{\circ}.\sin 40^{\circ}\\=\frac{1}{16}.\sin 20^{\circ}.\sin 40^{\circ}.\sin 60^{\circ}.\sin 80^{\circ}\\=\frac{\sqrt{3}}{32}.\sin 20^{\circ}.\sin 40^{\circ}.\sin 80^{\circ}=\frac{\sqrt{3}}{64}.\sin 20^{\circ}.\left( \cos 40^{\circ}-\cos 120^{\circ} \right)\\=\frac{\sqrt{3}}{64}.\sin 20^{\circ}.\left( \cos 40^{\circ}+\frac{1}{2} \right)=\frac{\sqrt{3}}{64}.\sin 20^{\circ}.\left( 2.\cos^{2}20^{\circ}-\frac{1}{2} \right)\\=\frac{\sqrt{3}}{128}.\sin 20^{\circ}.\left( 3-4.\sin^{2}20^{\circ} \right)=\frac{\sqrt{3}}{128}.\sin 60^{\circ}=\frac{3}{256}\end{array}$
d) Ta có
$K=96\sqrt{3}\sin\frac{\pi}{48}.\cos\frac{\pi}{48}.\cos\frac{\pi}{24}.\cos\frac{\pi}{12}.\cos\frac{\pi}{6}$
$\begin{array}{l}=48\sqrt{3}\sin\frac{\pi}{24}.\cos\frac{\pi}{24}.\cos\frac{\pi}{12}.\cos\frac{\pi}{6}\\=24\sqrt{3}\sin\frac{\pi}{12}.\cos\frac{\pi}{12}.\cos\frac{\pi}{6}\end{array}$
$=12\sqrt{3}\sin\frac{\pi}{6}.\cos\frac{\pi}{6}=6\sqrt{3}\sin\frac{\pi}{3}=9$
e) $L=\cos\frac{\pi}{15}.\cos\frac{2\pi}{15}.\cos\frac{3\pi}{15}.\cos\frac{4\pi}{15}.\cos\frac{5\pi}{15}.\cos\frac{6\pi}{15}.\cos\frac{7\pi}{15}$
$\begin{array}{l}=\cos\frac{\pi}{15}.\cos\frac{2\pi}{15}.\cos\frac{3\pi}{15}.\cos\frac{4\pi}{15}.\cos\frac{5\pi}{15}.\cos\frac{6\pi}{15}.\cos\frac{7\pi}{15}\\=-\frac{1}{2}.\left( \cos\frac{\pi}{15}.\cos\frac{2\pi}{15}.\cos\frac{4\pi}{15}.\cos\frac{8\pi}{15} \right).\left( \cos\frac{3\pi}{15}.\cos\frac{6\pi}{15} \right)\\=-\frac{1}{2}.\left( \cos\frac{\pi}{15}.\cos\left( 2.\frac{\pi}{15} \right).\cos\left( 2^{2}.\frac{\pi}{15} \right).\cos\left( 2^{3}.\frac{\pi}{15} \right) \right).\left( \cos\frac{3\pi}{15}.\cos\left( 2.\frac{3\pi}{15} \right) \right)\end{array}$
$\begin{array}{l}=-\frac{1}{2}.\left( \frac{\sin\left( 2^{4}.\frac{\pi}{15} \right)}{16.\sin\left( \frac{\pi}{15} \right)} \right).\left( \frac{\sin\left( 2^{2}.\frac{3\pi}{15} \right)}{4.\sin\left( \frac{3\pi}{15} \right)} \right)=-\frac{1}{2}.\left( \frac{\sin\left( \frac{16\pi}{15} \right)}{16.\sin\left( \frac{\pi}{15} \right)} \right).\left( \frac{\sin\left( \frac{12\pi}{15} \right)}{4.\sin\left( \frac{3\pi}{15} \right)} \right)\\=-\frac{1}{2}.\left( \frac{-\sin\left( \frac{\pi}{15} \right)}{16.\sin\left( \frac{\pi}{15} \right)} \right).\left( \frac{\sin\left( \frac{3\pi}{15} \right)}{4.\sin\left( \frac{3\pi}{15} \right)} \right)=\frac{1}{128}\end{array}$
f) Ta có
$M=\sin\frac{\pi}{16}.\cos\frac{\pi}{16}.\cos\frac{\pi}{8}=\frac{1}{2}\sin\frac{\pi}{8}.\cos\frac{\pi}{8}=\frac{1}{4}\sin\frac{\pi}{4}=\frac{\sqrt{2}}{8}$
$G=\cos\frac{2\pi}{31}.\cos\frac{4\pi}{31}.\cos\frac{8\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}$
$\begin{array}{l}\Leftrightarrow G.\sin\frac{2\pi}{31}=\sin\frac{2\pi}{31}.\cos\frac{2\pi}{31}.\cos\frac{4\pi}{31}.\cos\frac{8\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}\\\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{2}\sin\frac{4\pi}{31}.\cos\frac{4\pi}{31}.\cos\frac{8\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}\\\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{4}\sin\frac{8\pi}{31}.\cos\frac{8\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}\\\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{8}\sin\frac{16\pi}{31}.\cos\frac{16\pi}{31}.\cos\frac{32\pi}{31}\\\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{16}\sin\frac{32\pi}{31}.\cos\frac{32\pi}{31}\end{array}$
$\begin{array}{l}\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{32}\sin\frac{64\pi}{31}\\\Leftrightarrow G.\sin\frac{2\pi}{31}=\frac{1}{32}\sin\left( \frac{2\pi}{31}+2\pi \right)\\\Leftrightarrow G=\frac{1}{32}\end{array}$
b) Ta có
$\begin{array}{l}H=\sin 5^{\circ}.\sin 15^{\circ}.\sin 25^{\circ}\ldots\sin 75^{\circ}.\sin 85^{\circ}\\=\sin 5^{\circ}.\cos 5^{\circ}.\sin 15^{\circ}.\cos 15^{\circ}.\sin 25^{\circ}.\cos 25^{\circ}.\sin 35^{\circ}.\cos 35^{\circ}.\sin 45^{\circ}\\=\frac{\sqrt{2}}{32}\sin 10^{\circ}.\sin 30^{\circ}.\sin 50^{\circ}.\sin 70^{\circ}\end{array}$
$=\frac{\sqrt{2}}{32.\cos 10^{\circ}}.\cos 10^{\circ}.\sin 10^{\circ}.\sin 30^{\circ}.\sin 50^{\circ}.\cos 20^{\circ}$
$\begin{array}{l}=\frac{\sqrt{2}}{64.\cos 10^{\circ}}.\sin 20^{\circ}.\cos 20^{\circ}.\frac{1}{2}.\sin 50^{\circ}\\=\frac{\sqrt{2}}{256.\cos 10^{\circ}}.\sin 40^{\circ}.\cos 40^{\circ}\\=\frac{\sqrt{2}}{512.\cos 10^{\circ}}.\sin 80^{\circ}=\frac{\sqrt{2}}{512}\end{array}$
c) Ta có
$I=\cos 10^{\circ}.\cos 20^{\circ}.\cos 30^{\circ}\ldots\cos 70^{\circ}.\cos 80^{\circ}$
$\begin{array}{l}=\cos 10^{\circ}.\sin 10^{\circ}.\cos 20^{\circ}.\sin 20^{\circ}.\cos 30^{\circ}.\sin 30^{\circ}\cos 40^{\circ}.\sin 40^{\circ}\\=\frac{1}{16}.\sin 20^{\circ}.\sin 40^{\circ}.\sin 60^{\circ}.\sin 80^{\circ}\\=\frac{\sqrt{3}}{32}.\sin 20^{\circ}.\sin 40^{\circ}.\sin 80^{\circ}=\frac{\sqrt{3}}{64}.\sin 20^{\circ}.\left( \cos 40^{\circ}-\cos 120^{\circ} \right)\\=\frac{\sqrt{3}}{64}.\sin 20^{\circ}.\left( \cos 40^{\circ}+\frac{1}{2} \right)=\frac{\sqrt{3}}{64}.\sin 20^{\circ}.\left( 2.\cos^{2}20^{\circ}-\frac{1}{2} \right)\\=\frac{\sqrt{3}}{128}.\sin 20^{\circ}.\left( 3-4.\sin^{2}20^{\circ} \right)=\frac{\sqrt{3}}{128}.\sin 60^{\circ}=\frac{3}{256}\end{array}$
d) Ta có
$K=96\sqrt{3}\sin\frac{\pi}{48}.\cos\frac{\pi}{48}.\cos\frac{\pi}{24}.\cos\frac{\pi}{12}.\cos\frac{\pi}{6}$
$\begin{array}{l}=48\sqrt{3}\sin\frac{\pi}{24}.\cos\frac{\pi}{24}.\cos\frac{\pi}{12}.\cos\frac{\pi}{6}\\=24\sqrt{3}\sin\frac{\pi}{12}.\cos\frac{\pi}{12}.\cos\frac{\pi}{6}\end{array}$
$=12\sqrt{3}\sin\frac{\pi}{6}.\cos\frac{\pi}{6}=6\sqrt{3}\sin\frac{\pi}{3}=9$
e) $L=\cos\frac{\pi}{15}.\cos\frac{2\pi}{15}.\cos\frac{3\pi}{15}.\cos\frac{4\pi}{15}.\cos\frac{5\pi}{15}.\cos\frac{6\pi}{15}.\cos\frac{7\pi}{15}$
$\begin{array}{l}=\cos\frac{\pi}{15}.\cos\frac{2\pi}{15}.\cos\frac{3\pi}{15}.\cos\frac{4\pi}{15}.\cos\frac{5\pi}{15}.\cos\frac{6\pi}{15}.\cos\frac{7\pi}{15}\\=-\frac{1}{2}.\left( \cos\frac{\pi}{15}.\cos\frac{2\pi}{15}.\cos\frac{4\pi}{15}.\cos\frac{8\pi}{15} \right).\left( \cos\frac{3\pi}{15}.\cos\frac{6\pi}{15} \right)\\=-\frac{1}{2}.\left( \cos\frac{\pi}{15}.\cos\left( 2.\frac{\pi}{15} \right).\cos\left( 2^{2}.\frac{\pi}{15} \right).\cos\left( 2^{3}.\frac{\pi}{15} \right) \right).\left( \cos\frac{3\pi}{15}.\cos\left( 2.\frac{3\pi}{15} \right) \right)\end{array}$
$\begin{array}{l}=-\frac{1}{2}.\left( \frac{\sin\left( 2^{4}.\frac{\pi}{15} \right)}{16.\sin\left( \frac{\pi}{15} \right)} \right).\left( \frac{\sin\left( 2^{2}.\frac{3\pi}{15} \right)}{4.\sin\left( \frac{3\pi}{15} \right)} \right)=-\frac{1}{2}.\left( \frac{\sin\left( \frac{16\pi}{15} \right)}{16.\sin\left( \frac{\pi}{15} \right)} \right).\left( \frac{\sin\left( \frac{12\pi}{15} \right)}{4.\sin\left( \frac{3\pi}{15} \right)} \right)\\=-\frac{1}{2}.\left( \frac{-\sin\left( \frac{\pi}{15} \right)}{16.\sin\left( \frac{\pi}{15} \right)} \right).\left( \frac{\sin\left( \frac{3\pi}{15} \right)}{4.\sin\left( \frac{3\pi}{15} \right)} \right)=\frac{1}{128}\end{array}$
f) Ta có
$M=\sin\frac{\pi}{16}.\cos\frac{\pi}{16}.\cos\frac{\pi}{8}=\frac{1}{2}\sin\frac{\pi}{8}.\cos\frac{\pi}{8}=\frac{1}{4}\sin\frac{\pi}{4}=\frac{\sqrt{2}}{8}$