Trigonometric equations and inequalities

9000086707

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

9000086708

Level: 
A
Identify the equation which arises from the following equation using an optimal substitution. \[ \mathop{\mathrm{tg}}\nolimits ^{2}v -\mathop{\mathrm{cotg}}\nolimits ^{-1}v = 2 \]
\(t^{2} - t - 2 = 0\)
Equation is not convenient for a substitution.
\(t^{2} + t = 0\)
\(\mathop{\mathrm{tg}}\nolimits t = 2\)

9000046609

Level: 
B
Identify an inequality which is true for every \(x\) from the interval \(\left ( \frac{\pi }{4}; \frac{3\pi } {4}\right )\).
\(\sin x\geq \frac{\sqrt{2}} {2} \)
\(\mathop{\mathrm{tg}}\nolimits x > 1\)
\(\cos x > 0\)
\(\mathop{\mathrm{cotg}}\nolimits x\geq \frac{\sqrt{3}} {3} \)

9000046610

Level: 
B
Identify an inequality which is true for every \(x\) from the interval \(\left (\frac{5\pi } {6}; \frac{3\pi } {2}\right )\).
\(\cos x < \frac{1} {2}\)
\(\mathop{\mathrm{tg}}\nolimits x < 0\)
\(\sin x\geq -\frac{\sqrt{2}} {2} \)
\(\mathop{\mathrm{cotg}}\nolimits x < 1\)