# Test: Trigonometric equations I - Normal

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Question 1:   Solve the equation $\text{\hspace{0.17em}}\mathrm{cos}2x=-\frac{1}{2}\text{\hspace{0.17em}}$ .
$x=±\frac{\pi }{6}+2\pi n$
$x=±\frac{\pi }{6}+\pi n$
$x=±\frac{2\pi }{3}+\pi n$
$x=±\frac{\pi }{3}+\pi n$
Question 2:   Solve the equations $\text{\hspace{0.17em}}\mathrm{sin}\left(x-\frac{\pi }{6}\right)=-\frac{\sqrt{2}}{2}\text{\hspace{0.17em}}$ .
$x={\left(-1\right)}^{n+1}×\frac{\pi }{4}+\frac{\pi }{6}+\pi n,\text{\hspace{0.17em}}n\in Z$
$x={\left(-1\right)}^{n+1}×\frac{\pi }{4}-\frac{\pi }{6}+\pi n,\text{\hspace{0.17em}}n\in Z$
$x={\left(-1\right)}^{n}×\frac{\pi }{4}+\frac{\pi }{6}+\pi n,\text{\hspace{0.17em}}n\in Z$
$x={\left(-1\right)}^{n+1}×\frac{\pi }{4}-\frac{\pi }{6}+2\pi n,\text{\hspace{0.17em}}n\in Z$
Question 3:   Solve the equation $\text{\hspace{0.17em}}tg\left(-\frac{2x}{3}\right)=-\sqrt{3}\text{\hspace{0.17em}}$ .
$x=\frac{\pi }{2}+\frac{3\pi n}{2},\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{4}+\frac{3\pi n}{2},\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{3}+\frac{3\pi n}{2},\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{2}+\pi n,\text{\hspace{0.17em}}n\in Z$
Question 4:   Solve the equation $\text{\hspace{0.17em}}ctg6x=0\text{\hspace{0.17em}}$ .
$x=\frac{\pi }{12}+\frac{\pi n}{6},\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{12}+\pi n,\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{12}+\frac{\pi n}{12},\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{6}+\frac{\pi n}{6},\text{\hspace{0.17em}}n\in Z$
Question 5:   Solve the expression $\text{\hspace{0.17em}}\mathrm{sin}\left(\mathrm{arcsin}\frac{\pi }{3}\right)\text{\hspace{0.17em}}$ .
$\frac{1}{2}$
$\frac{\pi }{6}$
$\frac{\pi }{3}$
$\frac{\sqrt{3}}{2}$
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