Question

GPT

a) \(sinx=-\frac{\sqrt{3}}{2}\) voi \(x\in\left(0;2\pi\right)\)

b) \(2sin2x+1=0\) voi \(0< x< 90^o\)

c) \(2cos\left(x-\frac{\pi}{3}\right)=1\) voi \(-\pi< x< \pi\)

d) \(cos^3x-2cos^2x=0\) voi moi \(x\in\left[0;720^o\right]\)

Answer 1

a.

\(\Leftrightarrow\left[{}\begin{matrix}x=-\frac{\pi}{3}+k2\pi\\x=\frac{4\pi}{3}+k2\pi\end{matrix}\right.\)

\(\Rightarrow x=\left\{\frac{4\pi}{3};\frac{5\pi}{3}\right\}\)

b.

\(\Leftrightarrow sin2x=-\frac{1}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=-30^0+k360^0\\2x=210^0+k360^0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-15^0+k180^0\\x=105^0+k180^0\end{matrix}\right.\)

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Answer 2

c.

\(\Leftrightarrow cos\left(x-\frac{\pi}{3}\right)=\frac{1}{2}\)

\(\Leftrightarrow\left[{}\begin{matrix}x-\frac{\pi}{3}=\frac{\pi}{3}+k2\pi\\x-\frac{\pi}{3}=-\frac{\pi}{3}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{2\pi}{3}+k2\pi\\x=k2\pi\end{matrix}\right.\)

\(\Rightarrow x=\left\{0;\frac{2\pi}{3}\right\}\)

d.

\(\Leftrightarrow cos^2x\left(cosx-2\right)=0\)

\(\Leftrightarrow cosx=0\)

\(\Leftrightarrow x=90^0+k180^0\)

\(\Rightarrow x=\left\{90^0;270^0;450^0;630^0\right\}\)