Gọi $ {{x}_{0}} $ là nghiệm dương nhỏ nhất của phương trình $ 3si{{n}^{2}}x+2\sin x\cos x-co{{s}^{2}}x=0 $ . Chọn khẳng định đúng?A. $ {{x}_{0}}\in \left( \pi ;\dfrac{3\pi }{2} \right) $ .B. $ {{x}_{0}}\in \left( \dfrac{3\pi }{2};2\pi \right) $ .C. $ {{x}

trucquyen241

2 năm trước

Gọi $ {{x}_{0}} $ là nghiệm dương nhỏ nhất của phương trình $ 3si{{n}^{2}}x+2\sin x\cos x-co{{s}^{2}}x=0 $ . Chọn khẳng định đúng?
A. $ {{x}_{0}}\in \left( \pi ;\dfrac{3\pi }{2} \right) $ .
B. $ {{x}_{0}}\in \left( \dfrac{3\pi }{2};2\pi \right) $ .
C. $ {{x}_{0}}\in \left( \dfrac{\pi }{2};\pi \right) $ .
D. $ {{x}_{0}}\in \left( 0;\dfrac{\pi }{2} \right) $ .

Loga Hóa Học lớp 12

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minhphuc0102

Đáp án đúng: D
$ 3si{{n}^{2}}x+2\sin x\cos x-co{{s}^{2}}x=0\,\,\left( 1 \right) $ TH 1: $ \cos x=0 $ . PT $ \left( 1 \right) $ $ \Leftrightarrow 3{{\sin }^{2}}x=0\Leftrightarrow 3=0 $ vô nghiệm ( Vì $ \cos x=0\Leftrightarrow {{\sin }^{2}}x=1 $ ). TH 2: $ \cos x
e 0 $ . PT $ \left( 1 \right) $ $ \Leftrightarrow 3{{\tan }^{2}}x+2\tan x-1=0\Leftrightarrow \left[ \begin{array}{l} \operatorname{tanx}=-1 \\ \operatorname{tanx}=\dfrac{1}{3} \end{array} \right.\Leftrightarrow \left[ \begin{array}{l} x=-\dfrac{\pi }{4}+k\pi \\ x=\arctan \left( \dfrac{1}{3} \right)+k\pi \end{array} \right.\left( k\in \mathbb{Z} \right) $ . TH1: $ x=-\dfrac{\pi }{4}+k\pi > 0\Leftrightarrow k > \dfrac{1}{4}\Rightarrow {{k}_{\min }}=1\,\left( k\in \mathbb{Z} \right)\Rightarrow x=\dfrac{3\pi }{4} $ TH2: $ \begin{array}{l} x=\arctan \left( \dfrac{1}{3} \right)+k\pi > 0\Leftrightarrow k > -\dfrac{1}{\pi }.\arctan \left( \dfrac{1}{3} \right)\Rightarrow {{k}_{\min }}=0\,\,\left( k\in \mathbb{Z} \right) \\ \Rightarrow x=\arctan \left( \dfrac{1}{3} \right)\approx 0,32 \end{array} $ Vậy nghiệm dương nhỏ nhất của phương trình là: $ {{x}_{0}}=\arctan \left( \dfrac{1}{3} \right)\in \left( 0;\dfrac{\pi }{2} \right) $ .

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