tính tổng các nghiệm của ft $cos^{2}$ x – sin2x = $\sqrt[]{2}$ $cos^{2}$ ($\frac{\pi}{2}$ + x ) ∈ (0 2$\pi$ ) 26/08/2021 Bởi Sarah tính tổng các nghiệm của ft $cos^{2}$ x – sin2x = $\sqrt[]{2}$ $cos^{2}$ ($\frac{\pi}{2}$ + x ) ∈ (0 2$\pi$ )
Đáp án: $\begin{array}{l}{\cos ^2}x – \sin 2x = \sqrt 2 {\cos ^2}\left( {\frac{\pi }{2} + x} \right)\\ \Rightarrow {\cos ^2}x – 2.\sin x.\cos x = \sqrt 2 {\sin ^2}x\\ \Rightarrow {\cos ^2}x – 2\sin x.\cos x – \sqrt 2 {\sin ^2}x = 0\\ + Khi:\cos x = 0\\ \Rightarrow \sin x = 0\left( {ktm} \right)\\ + Khi:\cos x \ne 0\\ \Rightarrow 1 – 2\frac{{\sin x}}{{\cos x}} – \sqrt 2 .\frac{{{{\sin }^2}x}}{{{{\cos }^2}x}} = 0\\ \Rightarrow 1 – 2.\tan x – \sqrt 2 .{\tan ^2}x = 0\\ \Rightarrow \left[ \begin{array}{l}\tan x = 0,39\\\tan x = – 1,8\end{array} \right.\\ \Rightarrow \left[ \begin{array}{l}x = {21^0} + k{.180^0}\\x = {61^0} + k{.180^0}\end{array} \right.\\Do:x \in \left( {0;{{360}^0}} \right)\\ \Rightarrow \left[ \begin{array}{l}x = {21^0}\\x = {201^0}\\x = {61^0}\\x = {241^0}\end{array} \right.\end{array}$ Bình luận
Đáp án:
$\begin{array}{l}
{\cos ^2}x – \sin 2x = \sqrt 2 {\cos ^2}\left( {\frac{\pi }{2} + x} \right)\\
\Rightarrow {\cos ^2}x – 2.\sin x.\cos x = \sqrt 2 {\sin ^2}x\\
\Rightarrow {\cos ^2}x – 2\sin x.\cos x – \sqrt 2 {\sin ^2}x = 0\\
+ Khi:\cos x = 0\\
\Rightarrow \sin x = 0\left( {ktm} \right)\\
+ Khi:\cos x \ne 0\\
\Rightarrow 1 – 2\frac{{\sin x}}{{\cos x}} – \sqrt 2 .\frac{{{{\sin }^2}x}}{{{{\cos }^2}x}} = 0\\
\Rightarrow 1 – 2.\tan x – \sqrt 2 .{\tan ^2}x = 0\\
\Rightarrow \left[ \begin{array}{l}
\tan x = 0,39\\
\tan x = – 1,8
\end{array} \right.\\
\Rightarrow \left[ \begin{array}{l}
x = {21^0} + k{.180^0}\\
x = {61^0} + k{.180^0}
\end{array} \right.\\
Do:x \in \left( {0;{{360}^0}} \right)\\
\Rightarrow \left[ \begin{array}{l}
x = {21^0}\\
x = {201^0}\\
x = {61^0}\\
x = {241^0}
\end{array} \right.
\end{array}$