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Câu 20 trang 29 SGK Đại số và Giải tích 11 Nâng cao

Tìm nghiệm của các phương trình sau trên khoảng đã cho
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Tìm nghiệm của các phương trình sau trên khoảng đã cho

LG a

\(\tan \left( {2x - {{15}^0}} \right) = 1\) với \( - {180^0} < x < {90^0}\)

Lời giải chi tiết:

\(\tan \left( {2x-{{15}^0}} \right) = 1\) \( \Leftrightarrow 2x - {15^0} = {45^0} + k{180^0}\) \(\Leftrightarrow {\rm{ }}2x{\rm{ }} = {\rm{ }}{15^0} + {\rm{ }}{45^0} + {\rm{ }}k{180^0} \) \(\Leftrightarrow {\rm{ }}x{\rm{ }} = {\rm{ }}{30^0} + {\rm{ }}k{90^0}\) \( - {180^0} < {\rm{ }}{30^0} + {\rm{ }}k{90^0} < {\rm{ }}{90^0}\)

\(\begin{array}{l}
\Leftrightarrow - {210^0} < k{90^0} < {60^0}\\
\Leftrightarrow - \frac{7}{3} < k < \frac{2}{3}
\end{array}\)

\(\Leftrightarrow k \in \left\{ { - 2; - 1;0} \right\}\) (do k nguyên) Vậy các nghiệm của phương trình là \(x =  - {150^0},{\rm{ }}x{\rm{ }} =  - {60^0}\) và \(x{\rm{ }} = {\rm{ }}{30^0}\)

LG b

\(\cot 3x = - {1 \over {\sqrt 3 }}\,\text{ với }\, - {\pi \over 2} < x < 0.\)

Lời giải chi tiết:

\(\begin{array}{l}
\cot 3x = - \frac{1}{{\sqrt 3 }} = \cot \left( { - \frac{\pi }{3}} \right)\\
\Leftrightarrow 3x = - \frac{\pi }{3} + k\pi \\
\Leftrightarrow x = - \frac{\pi }{9} + \frac{{k\pi }}{3}
\end{array}\)

Do \(- {\pi \over 2} < x < 0\) nên:

\(\begin{array}{l}
- \frac{\pi }{2} < - \frac{\pi }{9} + \frac{{k\pi }}{3} < 0\\
\Leftrightarrow - \frac{{7\pi }}{{18}} < \frac{{k\pi }}{3} < \frac{\pi }{9}\\
\Leftrightarrow - \frac{{7}}{6} < k < \frac{1}{3}
\end{array}\)

Vì k nguyên nên \(k=-1, k=0\). Vậy các nghiệm của phương trình là \(x = - {{4\pi } \over 9}\,\text{ và }\,x = - {\pi \over 9}.\)

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