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Câu 40 trang 46 SGK Đại số và Giải tích 11 Nâng cao

Tìm các nghiệm của mỗi phương trình sau trong khoảng đã cho
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Tìm các nghiệm của mỗi phương trình sau trong khoảng đã cho (khi cần tính gần đúng thì tính chính xác đến \({1 \over {10}}\) giây)

LG a

\(2{\sin ^2}x - 3\cos x = 2,0^\circ \le x \le 360^\circ \)

Lời giải chi tiết:

\(2{\sin ^2}x - 3\cos x = 2\)

\(\begin{array}{l}
\Leftrightarrow 2\left( {1 - {{\cos }^2}x} \right) - 3\cos x - 2 = 0\\
\Leftrightarrow 2 - 2{\cos ^2}x - 3\cos x - 2 = 0\\
\Leftrightarrow - 2{\cos ^2}x - 3\cos x = 0\\
\Leftrightarrow 2{\cos ^2}x + 3\cos x = 0\\
\Leftrightarrow \cos x\left( {2\cos x + 3} \right) = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = 0\\
2\cos x + 3 = 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\cos x = 0\\
\cos x = - \frac{3}{2}\left( {loai} \right)
\end{array} \right.\\
\Leftrightarrow x = {90^0} + k{180^0},k \in Z\\
{0^0} \le x \le {360^0}\\
\Leftrightarrow {0^0} \le {90^0} + k{180^0} \le {360^0}\\
\Leftrightarrow - {90^0} \le k{180^0} \le {270^0}\\
\Leftrightarrow - \frac{1}{2} \le k \le \frac{3}{2}
\end{array}\)

Mà \(k \in Z \Rightarrow k \in \left\{ {0;1} \right\}\) +) Với k=0 thì \(x = {90^0}\) +) Với k=1 thì \(x = {270^0}\) Vậy với điều kiện \(0^0≤ x ≤ 360^0\), phương trình có hai nghiệm là \(x = 90^0\) và \(x = 270^0\).

LG b

\(\tan x + 2\cot x = 3,180^\circ \le x \le 360^\circ \)

Lời giải chi tiết:

ĐKXĐ : \(\sin x ≠ 0\) và \(\cos x ≠ 0\). Ta có :

\(\begin{array}{l}
\tan x + 2\cot x = 3\\
\Leftrightarrow \tan x + \frac{2}{{\tan x}} - 3 = 0\\
\Leftrightarrow \frac{{{{\tan }^2}x + 2 - 3\tan x}}{{\tan x}} = 0\\
\Rightarrow {\tan ^2}x - 3\tan x + 2 = 0\\
\Leftrightarrow \left[ \begin{array}{l}
\tan x = 1\\
\tan x = 2
\end{array} \right.
\end{array}\)

+) \( \tan x = 1 ⇔ x = 45^0 + k180^0\).

\(\begin{array}{l}
{180^0} \le x \le {360^0}\\
\Rightarrow {180^0} \le {45^0} + k{180^0} \le {360^0}\\
\Leftrightarrow {135^0} \le k{180^0} \le {315^0}\\
\Leftrightarrow \frac{3}{4} \le k \le \frac{7}{4} \Rightarrow k = 1
\end{array}\)

Có một nghiệm thỏa mãn \(180^0\le {\rm{ }}x{\rm{ }} \le {\rm{ }}360^0\), ứng với \(k = 1\) là \(x = 225^0\) +) \( \tan x = 2 ⇔ x = α + k180^0\) với \(\tan α = 2\). Ta có thể chọn \(\alpha  \approx {63^0}26'\)

\(\begin{array}{l}
{180^0} \le x \le {360^0}\\
\Rightarrow {180^0} \le {63^0}26' + k{180^0} \le {360^0}\\
\Leftrightarrow {116^0}34' \le k{180^0} \le {296^0}34'\\
\Leftrightarrow 0,64 < k < 1,65 \Rightarrow k = 1
\end{array}\)

Vậy có một nghiệm (gần đúng) thỏa mãn \(180^0\le {\rm{ }}x{\rm{ }} \le {\rm{ }}360^0\) là : \(x = \alpha  + {180^0} \approx {243^0}26'\) Kết luận : Với điều kiện \(180^0\le {\rm{ }}x{\rm{ }} \le {\rm{ }}360^0\), phương trình có hai nghiệm \(x = 225^0\) và \(x \approx {243^0}26'\).

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