Bài tập lượng giác

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Bài 1: Rút gọn:
a) [TEX]T1= \frac{1}{sin a} + \frac{1}{sin 2a} + \frac{1}{sin 4a} +...+ \frac{1}{sin 2^n .a}[/TEX]

b) [TEX]T2= sin^2 \frac{\pi}{13} + sin^2 \frac{3. \pi}{13} + sin^2 \frac{5. \pi}{13} +sin^2 \frac{7. \pi}{13} + sin^2 \frac{9. \pi}{13} [/TEX]

Bài 2: Tính tổng

[TEX]S=\frac{1}{cos (\frac{\pi}{7}) } + \frac{1}{cos (\frac{3 \pi}{7}) } + \frac{1}{cos (\frac{5 \pi}{7}) }[/TEX]

 

[lp_qt]

Câu 2

•$$\dfrac{3\pi }{7}+\dfrac{4\pi }{7}=\pi \rightarrow cos\dfrac{3\pi }{7}+cos\dfrac{4\pi }{7}=0 (1)$$
Có : $$cos\dfrac{3\pi }{7}=4cos^3\dfrac{\pi }{7}-3cos\dfrac{\pi }{7}$$
$$cos\dfrac{4\pi }{7}=2cos^2\dfrac{2\pi }{7}-1=2(2cos\dfrac{\pi }{7}-1)^2-1$$
Đặt $$ a=cos\dfrac{\pi }{7} (0<a<1)\rightarrow 4a^3-3a+2(2a-1)^2-1=0$$
$$\leftrightarrow \begin{bmatrix}& a=-1(ktm) & \\ & 8a^3-4a^2-4a+1=0 & \end{bmatrix}\rightarrow 8a^3-4a^2-4a+1=0$$

•$$3.\dfrac{3\pi }{7}+4.\dfrac{3\pi }{7}=3\pi \leftrightarrow \dfrac{9\pi }{7}+\dfrac{12\pi }{7}=3\pi$$
$$\rightarrow cos\dfrac{9\pi }{7}+cos\dfrac{12\pi }{7}=0 (2)$$
Đặt $$b=cos\dfrac{3\pi }{7} (0<a<1)$$
$$TT \rightarrow PT :8b^3-4b^2-4b+1=0$$

• $$3.\dfrac{5\pi }{7}+4.\dfrac{5\pi }{7}=5\pi \leftrightarrow \dfrac{15\pi }{7}+\dfrac{12\pi }{7}=5\pi$$
$$\rightarrow cos\dfrac{15\pi }{7}+cos\dfrac{20\pi }{7}=0 (3)$$
Đặt $$c=cos\dfrac{5\pi }{7} (0<a<1)$$
$$TT \rightarrow PT :8c^3-4c^2-4c+1=0$$

Vậy : $cos\dfrac{\pi }{7} ;cos\dfrac{3\pi }{7} ;cos\dfrac{5\pi }{7} $ là 3 nghiệm của phương trình: $$ 8x^3-4x^2-4x+1=0$$

Theo Định lý Vi-ét đối với phương trình bậc ba, ta có:
$$\left\{\begin{matrix}
cos\dfrac{\pi }{7} . cos\dfrac{3\pi }{7} .cos\dfrac{5\pi }{7} =\dfrac{-
1}{8}& \\
cos\dfrac{\pi }{7} . cos\dfrac{3\pi }{7} +cos\dfrac{\pi }{7} . cos\dfrac{5\pi }{7} +cos\dfrac{3\pi }{7} . cos\dfrac{5\pi }{7} =\dfrac{-1}{2} &
\end{matrix}\right.$$

Ta có:

$$S=\dfrac{1}{cos\dfrac{\pi}{7}}+ \dfrac{1}{cos\dfrac{3\pi}{7}}+\dfrac{1}{cos\dfrac{5\pi}{7}}=\dfrac{cos\dfrac{\pi }{7} . cos\dfrac{3\pi }{7} +cos\dfrac{\pi }{7} . cos\dfrac{5\pi }{7} +cos\dfrac{3\pi }{7} . cos\dfrac{5\pi }{7} }{cos\dfrac{\pi }{7} . cos\dfrac{3\pi }{7} .cos\dfrac{5\pi }{7} }=....$$

 

[lp_qt]

$$T= sin^2 \left ( \dfrac{\pi}{13} \right ) + sin^2 \left ( \dfrac{3\pi}{13} \right ) + sin^2 \left ( \dfrac{5\pi}{13} \right )+ sin^2 \left (\dfrac{7\pi}{13} \right ) + sin^2 \left ( \dfrac{9\pi}{13} \right )$$

$$=\dfrac{1-cos \dfrac{2\pi}{13}}{2}+\dfrac{1-cos \dfrac{6\pi}{13}}{2}+\dfrac{1-cos \dfrac{10\pi}{13}}{2}+\dfrac{1-cos \dfrac{14\pi}{13}}{2}+\dfrac{1-cos \dfrac{18\pi}{13}}{2}$$

$$=\dfrac{5}{2}-\left ( cos\dfrac{2\pi}{13}+cos\dfrac{6\pi}{13}+cos\dfrac{10\pi}{13}+cos\dfrac{14\pi}{13}+cos\dfrac{18\pi}{13} \right )$$

$$P=cos\dfrac{2\pi}{13}+ cos\dfrac{6\pi}{13}+cos\dfrac{10\pi}{13}+cos\dfrac{14\pi}{13}+ cos\dfrac{18\pi}{13}$$

$$P.2.sin\dfrac{2\pi}{13}=2.sin\dfrac{2\pi}{13}.cos\dfrac{2\pi}{13}+2.sin\dfrac{2\pi}{13}.cos\dfrac{6\pi}{13}+2.sin\dfrac{2\pi}{13}.cos\dfrac{10\pi}{13}+2.sin\dfrac{2\pi}{13}.cos\dfrac{14\pi}{13}+2.sin\dfrac{2\pi}{13}.cos\dfrac{18\pi}{13}$$

$$=\left ( sin0 + sin\dfrac{4\pi}{13} \right )+\left ( sin\dfrac{-4\pi}{13}+sin\dfrac{8\pi}{13} \right )+\left ( sin\dfrac{-8\pi}{13}+sin\dfrac{12\pi}{13} \right )+\left (sin\dfrac{-12\pi}{13}+sin\dfrac{16\pi}{13} \right )+\left (sin\dfrac{-16\pi}{13}+sin\dfrac{20\pi}{13} \right )$$

$$=sin\dfrac{20\pi}{13}$$

$$\rightarrow P=\dfrac{sin\dfrac{20\pi}{13} }{2.sin\dfrac{2\pi}{13}}$$

$$\rightarrow T=\dfrac{5}{2}-\dfrac{sin\dfrac{20\pi}{13} }{2.sin\dfrac{2\pi}{13}}$$