[Toán 10] Lượng giác

L

lp_qt

cosA+cosB+cosC+cosπ3cosA+cosB+cosC+cos\dfrac{\pi }{3}

=2.cosA+B2.cosAB2+2.cos(C2+π6).cos(C2π6)=2.cos\dfrac{A+B}{2}.cos\dfrac{A-B}{2}+2.cos(\dfrac{C}{2}+\dfrac{\pi }{6}).cos(\dfrac{C}{2}-\dfrac{\pi }{6})

2.cosA+B2+2.cos(C2+π6)=4.cos(A+B+C12+π12).cos(A+B+C12π12)\le 2.cos\dfrac{A+B}{2}+2.cos(\dfrac{C}{2}+\dfrac{\pi }{6})=4.cos(\dfrac{A+B+C}{12}+\dfrac{\pi }{12}).cos(\dfrac{A+B+C}{12}-\dfrac{\pi }{12})

4.cos(A+B+C12+π12)=4.cosπ3\le 4.cos(\dfrac{A+B+C}{12}+\dfrac{\pi }{12})=4.cos\dfrac{\pi }{3}

\Rightarrow cosA+cosB+cosC+cos3.cosπ3=32cosA+cosB+cosC+cos \le 3.cos\dfrac{\pi }{3}=\dfrac{3}{2}

dấu = xảy ra khi A=B=CA=B=C
 
L

lp_qt

cách khác

cosA+cosB+cosC32cosA+cosB+cosC \le \dfrac{3}{2}

\Leftrightarrow 2.cosA+B2.cosAB2+12.sin2C2+3202.cos\dfrac{A+B}{2}.cos\dfrac{A-B}{2}+1-2.sin^2\dfrac{C}{2}+\dfrac{3}{2} \le 0

\Leftrightarrow 2.sin2C2+2.sinC2.cosAB2120-2.sin^2\dfrac{C}{2}+2.sin\dfrac{C}{2}.cos\dfrac{A-B}{2}-\dfrac{1}{2} \le 0

f(t)=2t2+2t.cosAB212f(t)=-2t^2+2t.cos\dfrac{A-B}{2}-\dfrac{1}{2}

Δ=cos2AB210\Delta '=cos^2\dfrac{A-B}{2}-1\le 0

\Rightarrow f(t)0f(t) \le 0
\Rightarrow đpcm
 
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