Trigonometric equations and inequalities

2000006402

Level: 
A
An equation is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding equation.
\[ \mathrm{tg}\,{x} = {\sqrt{3}} \] \[ x \in [ 0 ;2\pi]\]
\[ \mathrm{tg}\,{x} = {\sqrt{3}} \] \[ x \in [ -\pi ;\pi]\]
\[ \mathrm{cotg}\,{x} = {\sqrt{3}} \] \[ x \in [ 0 ;2\pi]\]
\[ \mathrm{cotg}\,{x} = {\sqrt{3}} \] \[ x \in [ -\pi ;\pi]\]

2000006403

Level: 
A
An equation is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding equation.
\[ \mathrm{cotg}\,{x} = -\frac{\sqrt{3}}{3} \] \[ x \in ( -\pi ;2\pi)\]
\[ \mathrm{cotg}\,{x} = -\frac{1}{2} \] \[ x \in (-\pi ;2\pi )\]
\[ \mathrm{tg}\,{x} = -\frac{\sqrt{3}}{2} \] \[ x \in ( -\pi ;2\pi)\]
\[ \mathrm{tg}\,{x} = -\frac{1}{2} \] \[ x \in ( -\pi ;2\pi)\]

2000006404

Level: 
A
An equation is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding equation.
\[ \mathrm{cotg}\,{x} = 1\] \[ x \in ( -\pi ;2\pi)\]
\[ \mathrm{cotg}\,{x} = 1\] \[ x \in (0 ;2\pi )\]
\[ \mathrm{cotg}\,{x} = \frac{\sqrt{3}}{2} \] \[ x \in ( -\pi ;2\pi)\]
\[ \mathrm{cotg}\,{x} = \frac{\sqrt{3}}{2} \] \[ x \in ( 0 ;2\pi)\]

2010010701

Level: 
A
The solution set of the equation \( \cos x =-0.5 \) for \( x\in[ 0;2\pi ]\) is:
\( \left\{ \frac{2\pi}3;\frac{4\pi}3\right\} \)
\( \left\{ \frac{2\pi}3;\frac{5\pi}3\right\} \)
\( \left\{ \frac{4\pi}3;\frac{5\pi}3\right\} \)
\( \left\{ \frac{4\pi}3;\frac{7\pi}3\right\} \)

2010010702

Level: 
A
The solution set of the equation \( \mathrm{cotg}\, x =\sqrt{3} \) for \( x\in (-\pi;\pi )\) is:
\( \left\{ -\frac{5\pi}6;\frac{\pi}6\right\} \)
\( \left\{ -\frac{\pi}6;\frac{\pi}6\right\} \)
\( \left\{ -\frac{\pi}3;\frac{\pi}3\right\} \)
\( \left\{ -\frac{2\pi}3;\frac{\pi}3\right\} \)

2010010704

Level: 
A
Identify the equation which arises from the following equation using an optimal substitution. \[ \mathop{\mathrm{tg}}\nolimits x + \frac{2\sqrt{3}}{3}=\mathop{\mathrm{cotg}}\nolimits x \]
\(\sqrt{3}t^{2} +2t -\sqrt{3}= 0\)
\(t^{2} +2\sqrt{3}t-1= 0\)
\(3t^{2} -2\sqrt{3}t +{3}= 0\)
\(\sqrt{3}t^{2} +t +2\sqrt{3}= 0\)