# Test: Trigonometric equations I - Normal

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Question 1:   Solve the equation $\text{\hspace{0.17em}}\mathrm{cos}2x=-\frac{1}{2}\text{\hspace{0.17em}}$ .
$x=±\frac{2\pi }{3}+\pi n$
$x=±\frac{\pi }{6}+2\pi n$
$x=±\frac{\pi }{6}+\pi n$
$x=±\frac{\pi }{3}+\pi n$
Question 2:   Solve the equations $\text{\hspace{0.17em}}\mathrm{sin}\left(x-\frac{\pi }{6}\right)=-\frac{\sqrt{2}}{2}\text{\hspace{0.17em}}$ .
$x={\left(-1\right)}^{n+1}×\frac{\pi }{4}+\frac{\pi }{6}+\pi n,\text{\hspace{0.17em}}n\in Z$
$x={\left(-1\right)}^{n}×\frac{\pi }{4}+\frac{\pi }{6}+\pi n,\text{\hspace{0.17em}}n\in Z$
$x={\left(-1\right)}^{n+1}×\frac{\pi }{4}-\frac{\pi }{6}+\pi n,\text{\hspace{0.17em}}n\in Z$
$x={\left(-1\right)}^{n+1}×\frac{\pi }{4}-\frac{\pi }{6}+2\pi n,\text{\hspace{0.17em}}n\in Z$
Question 3:   Solve the equation $\text{\hspace{0.17em}}tg\left(-\frac{2x}{3}\right)=-\sqrt{3}\text{\hspace{0.17em}}$ .
$x=\frac{\pi }{3}+\frac{3\pi n}{2},\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{2}+\frac{3\pi n}{2},\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{4}+\frac{3\pi n}{2},\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{2}+\pi n,\text{\hspace{0.17em}}n\in Z$
Question 4:   Solve the equation $\text{\hspace{0.17em}}ctg6x=0\text{\hspace{0.17em}}$ .
$x=\frac{\pi }{6}+\frac{\pi n}{6},\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{12}+\frac{\pi n}{6},\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{12}+\frac{\pi n}{12},\text{\hspace{0.17em}}n\in Z$
$x=\frac{\pi }{12}+\pi n,\text{\hspace{0.17em}}n\in Z$
Question 5:   Solve the expression $\text{\hspace{0.17em}}\mathrm{sin}\left(\mathrm{arcsin}\frac{\pi }{3}\right)\text{\hspace{0.17em}}$ .
$\frac{\pi }{6}$
$\frac{1}{2}$
$\frac{\pi }{3}$
$\frac{\sqrt{3}}{2}$
Question 6:   Solve the expression $\text{\hspace{0.17em}}\mathrm{cos}\left(2arctg\left(-\frac{1}{3}\right)\right)\text{\hspace{0.17em}}$ .
$\frac{2\sqrt{3}}{3}$
$\frac{4}{5}$
$-\frac{4}{5}$
$\frac{11}{9}$
Question 7:   Simplify the expression $\text{\hspace{0.17em}}{\mathrm{sin}}^{2}4\alpha -{\mathrm{cos}}^{2}4\alpha \text{\hspace{0.17em}}$ .
$-\mathrm{cos}4\alpha$
$-\mathrm{cos}8\alpha$
$\mathrm{cos}8\alpha$
$1$
Question 8:   Simplify the expression $\text{\hspace{0.17em}}{\mathrm{cos}}^{2}6\beta +\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\left(-6\beta \right)\text{\hspace{0.17em}}$ .
$\mathrm{cos}2\beta$
-$\mathrm{cos}2\beta$
$1$
$-1$
Question 9:   Solve the equation $\text{\hspace{0.17em}}\mathrm{sin}2x+\mathrm{cos}x=0\text{\hspace{0.17em}}$ .
$x={\left(-1\right)}^{n+1}×\frac{\pi }{6}+\pi n,\text{\hspace{0.17em}}n\in Z$
$\begin{array}{l}{x}_{1}=\frac{\pi }{2}+\pi n,\text{\hspace{0.17em}}n\in Z\\ {x}_{2}={\left(-1\right)}^{n+1}×\frac{\pi }{6}+\pi n,\text{\hspace{0.17em}}n\in Z\end{array}$
$\varnothing$
$x=\frac{\pi }{2}+\pi n,\text{\hspace{0.17em}}n\in Z$
Question 10:   Solve the equation $\text{\hspace{0.17em}}\frac{tgx+1}{{\mathrm{cos}}^{2}x}=3\text{\hspace{0.17em}}$ .
$\begin{array}{l}{x}_{1}=1\text{\hspace{0.17em}}\\ {x}_{2}=-2\end{array}$
$\begin{array}{l}{x}_{1}=-1\text{\hspace{0.17em}}\\ {x}_{2}=2\end{array}$
$0$
$\varnothing$