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a.
\(\left\{{}\begin{matrix}sin\left(3x+\dfrac{\pi}{6}\right)\ne0\\cos2x\ne0\\sinx\ne-1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\ne-\dfrac{\pi}{18}+\dfrac{k\pi}{3}\\x\ne\dfrac{\pi}{4}+\dfrac{k\pi}{2}\\x\ne-\dfrac{\pi}{2}+k2\pi\end{matrix}\right.\)
b.
Do \(5+2cot^2x-sinx=4+2cot^2x+\left(1-sinx\right)>0\) nên hàm xác định khi:
\(\left\{{}\begin{matrix}sinx\ne0\\sin\left(x+\dfrac{\pi}{2}\right)\ne0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}sinx\ne0\\cosx\ne0\end{matrix}\right.\) \(\Leftrightarrow sin2x\ne0\)
\(\Leftrightarrow x\ne\dfrac{k\pi}{2}\)
1.
ĐKXĐ: \(\left\{{}\begin{matrix}cosx\ne0\\tanx-sinx\ne0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}cosx\ne0\\\dfrac{sinx}{cosx}-sinx\ne0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}cosx\ne0\\sinx\ne0\\cosx\ne1\end{matrix}\right.\) \(\Leftrightarrow sin2x\ne0\Leftrightarrow x\ne\dfrac{k\pi}{2}\)
2.
ĐKXĐ: \(sin2x\ne0\Leftrightarrow x\ne\dfrac{k\pi}{2}\)
3.
ĐKXĐ: \(\left\{{}\begin{matrix}sin\left(x-\dfrac{\pi}{4}\right)\ne0\\cos\left(x-\dfrac{\pi}{4}\right)\ne0\end{matrix}\right.\)
\(\Leftrightarrow sin\left(2x-\dfrac{\pi}{2}\right)\ne0\Leftrightarrow cos2x\ne0\)
\(\Leftrightarrow x\ne\dfrac{\pi}{4}+\dfrac{k\pi}{2}\)
1. \(sin\left(\dfrac{\pi}{3}-x\right)\ne0\Leftrightarrow\dfrac{\pi}{3}-x\ne k\pi\Leftrightarrow x\ne\dfrac{\pi}{3}-k\pi\)
2. \(cos2x\ne0\Leftrightarrow2x\ne\dfrac{\pi}{2}+k\pi\Leftrightarrow x\ne\dfrac{\pi}{4}+\dfrac{k\pi}{2}\)
3. \(\sqrt{1+sinx}-\sqrt{2}\ge0\Leftrightarrow1+sinx\ge2\Leftrightarrow sinx\ge1\Leftrightarrow sinx=1\Leftrightarrow x=\dfrac{\pi}{2}+k2\pi\)
4. \(\sqrt{2-2cosx}-2\ne0\Leftrightarrow2-2cosx\ne4\Leftrightarrow cosx\ne-1\Leftrightarrow x\ne\pi+k2\pi\)
5. \(1-\sqrt{1+sin3x}\ne0\Leftrightarrow sin3x\ne0\Leftrightarrow3x\ne k\pi\Leftrightarrow x\ne\dfrac{k\pi}{3}\)
ĐKXĐ:
a.
\(sin5x\ne0\Leftrightarrow5x\ne k\pi\Rightarrow x\ne\dfrac{k\pi}{5}\)
b.
\(cos6x\ne0\Leftrightarrow6x\ne\dfrac{\pi}{2}+k\pi\Rightarrow x\ne\dfrac{\pi}{12}+\dfrac{k\pi}{6}\)
d.
\(sin\left(3x-\dfrac{\pi}{6}\right)\ne0\Leftrightarrow3x-\dfrac{\pi}{6}\ne k\pi\Rightarrow x\ne\dfrac{\pi}{18}+\dfrac{k\pi}{3}\)
e.
\(sin\left(4x-\dfrac{\pi}{3}\right)\ne0\Leftrightarrow4x-\dfrac{\pi}{3}\ne k\pi\Rightarrow x\ne\dfrac{\pi}{12}+\dfrac{k\pi}{4}\)
1. \(D=R\)
2. \(sinx\ne0\Leftrightarrow x\ne k\pi\Rightarrow D=R\backslash\left\{k\pi|k\in R\right\}\)
3. \(cos2x\ne0\Leftrightarrow2x\ne\dfrac{\pi}{2}+k\pi\Leftrightarrow x\ne\dfrac{\pi}{4}+\dfrac{k\pi}{2}\Rightarrow D=R\backslash\left\{\dfrac{\pi}{4}+\dfrac{k\pi}{2}|k\in R\right\}\)
4. \(cos\left(x+\dfrac{\pi}{4}\right)\ne0\Leftrightarrow x+\dfrac{\pi}{4}\ne\dfrac{\pi}{2}+k\pi\Leftrightarrow x\ne\dfrac{\pi}{4}+k\pi\Rightarrow D=R\backslash\left\{\dfrac{\pi}{4}+k\pi|k\in R\right\}\)
1.
ĐKXĐ: \(sin\left(2x+3\right)\ne0\Leftrightarrow2x+3\ne k\pi\)
\(\Leftrightarrow x\ne-\frac{3}{2}+\frac{k\pi}{2}\)
2.
ĐKXĐ: \(\left\{{}\begin{matrix}cos2x\ne0\\sinx\ne-1\\sin\left(3x+\frac{\pi}{6}\right)\ne0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}2x\ne\frac{\pi}{2}+k\pi\\x\ne-\frac{\pi}{2}+k2\pi\\3x+\frac{\pi}{6}\ne k\pi\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ne\frac{\pi}{4}+\frac{k\pi}{2}\\x\ne-\frac{\pi}{2}+k2\pi\\x\ne-\frac{\pi}{18}+\frac{k\pi}{3}\end{matrix}\right.\)
3.
\(\left\{{}\begin{matrix}cos5x\ne0\\sin4x\ne cos3x\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}5x\ne\frac{\pi}{2}+k\pi\\sin4x\ne sin\left(\frac{\pi}{2}-3x\right)\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ne\frac{\pi}{10}+\frac{k\pi}{5}\\4x\ne\frac{\pi}{2}-3x+k2\pi\\4x\ne\frac{\pi}{2}+3x+k2\pi\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ne\frac{\pi}{10}+\frac{k\pi}{5}\\x\ne\frac{\pi}{14}+\frac{k2\pi}{7}\\x\ne\frac{\pi}{2}+k2\pi\end{matrix}\right.\)