Question #190734

Provide the values of β that are the solutions to the equation:

2cos(3β + π/3) = -√2



1
Expert's answer
2021-05-19T14:19:42-0400

Provide the values of β that are the solutions to the equation:

2cos(3β+π3)=22\cos\left(3\beta+\frac{\pi}{3}\right)=-\sqrt2.

Solution:\textbf{Solution:}

2cos(3β+π3)=22\cos\left(3\beta+\frac{\pi}{3}\right)=-\sqrt2.


[3β+π3=cos1(22)+2πk,kZ,3β+π3=cos1(22)+2πk,kZ.\left[ \begin{gathered} 3\beta+\frac{\pi}{3}=\cos^{-1}\left(-\frac{\sqrt2}{2}\right)+2\pi k, \, k\in\mathbb {Z}, \\ 3\beta+\frac{\pi}{3}=-\cos^{-1}\left(-\frac{\sqrt2}{2}\right)+2\pi k, \, k\in\mathbb {Z}. \end{gathered} \right.

[3β+π3=3π4+2πk,kZ,3β+π3=3π4+2πk,kZ.\left[ \begin{gathered} 3\beta+\frac{\pi}{3}=\frac{3\pi}{4}+2\pi k, \, k\in\mathbb {Z}, \\ 3\beta+\frac{\pi}{3}=-\frac{3\pi}{4}+2\pi k, \, k\in\mathbb {Z}. \end{gathered} \right.

[3β=5π12+2πk,kZ,3β=13π12+2πk,kZ.\left[ \begin{gathered} 3\beta=\frac{5\pi}{12}+2\pi k, \, k\in\mathbb {Z}, \\ 3\beta=-\frac{13\pi}{12}+2\pi k, \, k\in\mathbb {Z}. \end{gathered} \right.

[β=5π36+2πk3,kZ,β=13π36+2πk3,kZ.\left[ \begin{gathered} \beta=\frac{5\pi}{36}+\frac{2\pi k}{3}, \, k\in\mathbb {Z}, \\ \beta=-\frac{13\pi}{36}+\frac{2\pi k}{3}, \, k\in\mathbb {Z}. \end{gathered} \right.

Answer:\textbf{Answer:}

β{5π36+2πk3}{13π36+2πk3},kZ\boxed{\beta\in \{\frac{5\pi}{36}+\frac{2\pi k}{3} \}\cup \{-\frac{13\pi}{36}+\frac{2\pi k}{3}\}, \, k\in\mathbb {Z}}


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