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Tìm 1 phương trình bậc 3 có nghiệm là : $cos\dfrac{\pi}{7};cos\dfrac{3\pi}{7};cos\dfrac{5\pi}{7}$
♦$$\dfrac{3\pi }{7}+\dfrac{4\pi }{7}=\pi \rightarrow cos\dfrac{3\pi }{7}+cos\dfrac{4\pi }{7}=0 (1)$$
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 $$c=cos\dfrac{5\pi }{7} (0<a<1)$$
$$TT \rightarrow PT :8c^3-4c^2-4c+1=0$$
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