9000046603
Level:
B
Identify an inequality which is true for
\(x = \frac{23\pi }
{12}\).
\(\cos x > \frac{1}
{2}\)
\(\mathop{\mathrm{tg}}\nolimits x > 0\)
\(\cos x < \frac{\sqrt{2}}
{2} \)
\(\mathop{\mathrm{cotg}}\nolimits x > -1\)
Trigonometric equations and inequalities
9000046604
Level:
B
Identify an inequality which is true for
\(x = -\frac{5\pi }
{8}\).
\(\mathop{\mathrm{tg}}\nolimits x\geq \frac{\sqrt{3}}
{3} \)
\(\mathop{\mathrm{cotg}}\nolimits x > 1\)
\(\cos x > \frac{1}
{2}\)
\(\sin x > -\frac{1}
{2}\)
9000046605
Level:
B
Identify an inequality which is true for
\(x = \frac{\pi } {6}\).
\(\sin x\cdot \cos x < \frac{\sqrt{2}}
{2} \)
\(\cos 2x > \frac{\sqrt{2}}
{2} \)
\(\mathop{\mathrm{tg}}\nolimits (-x) > 0\)
\(\mathop{\mathrm{cotg}}\nolimits ^{2}x < 0\)
9000046606
Level:
B
Identify an inequality which is true for
\(x = \frac{3\pi }
{4}\).
\(\mathop{\mathrm{cotg}}\nolimits (-x) > 0\)
\(\sin 2x > 0\)
\(\mathop{\mathrm{tg}}\nolimits x\cdot \mathop{\mathrm{cotg}}\nolimits x < \frac{\sqrt{2}}
{2} \)
\(\cos ^{2}x < 0\)
9000046501
Level:
B
Identify the optimal first step convenient to solve the following trigonometric
equation. Do not consider the step which is possible but does not help to solve the
equation.
\[
\sin x\cdot \cos x = 0
\]
\(\sin 2x = 0\)
\(\cos 2x = 0\)
substitution \( \sin x = z\)
\(\sin ^{2}x\cdot \cos ^{2}x = 0\)
9000046503
Level:
A
Identify the optimal first step convenient to solve the following trigonometric
equation. Do not consider the step which is possible but does not help to solve the
equation.
\[
\mathop{\mathrm{tg}}\nolimits \left (-x + \frac{\pi }
{6}\right ) = \sqrt{3}
\]
substitution \( - x + \frac{\pi } {6} = z\)
\(\mathop{\mathrm{tg}}\nolimits (-x) = \sqrt{3} - \frac{\pi } {6}\)
\(\mathop{\mathrm{tg}}\nolimits ^{2}\left (-x + \frac{\pi } {6}\right ) = 3\)
\(\frac{\sin \left (-x+ \frac{\pi }{6} \right )}
{\cos \left (-x+ \frac{\pi }{6} \right )} = \sqrt{3}\)
9000046504
Level:
A
Identify the optimal first step convenient to solve the following trigonometric
equation. Do not consider the step which is possible but does not help to solve the
equation.
\[
\cos \left (x + \frac{\pi }
{3}\right ) = \frac{\sqrt{3}}
{2}
\]
substitution \( x + \frac{\pi } {3} = z\)
\(\cos ^{2}\left (x + \frac{\pi } {3}\right ) = \frac{3}
{4}\)
substitution \( \frac{\sqrt{3}}
{2} = z\)
\(\cos x\cdot \cos \frac{\pi }{3} -\sin x\cdot \sin \frac{\pi }{3} = \frac{\sqrt{3}}
{2} \)
9000046505
Level:
C
Identify the optimal first step convenient to solve the following trigonometric
equation. Do not consider the step which is possible but does not help to solve the
equation.
\[
\sin x = 1 +\cos x
\]
\(\sin ^{2}x = 1 + 2\cos x +\cos ^{2}x\)
\(\sin ^{2}x = 1 +\cos ^{2}x\)
substitution \( 1 +\cos x = z\)
\(\sin x -\cos x = z\)
9000046507
Level:
C
Identify the optimal first step convenient to solve the following trigonometric
equation. Do not consider the step which is possible but does not help to solve the
equation.
\[
\sqrt{3}\cos x = 1 -\sin x
\]
\(3\cos ^{2}x = (1 -\sin x)^{2}\)
\(3\cos ^{2}x = 1 -\sin ^{2}x\)
substitution \( 1 -\sin x = z\)
substitution \( \cos x = z\)
9000046508
Level:
C
Identify the optimal first step convenient to solve the following trigonometric
equation. Do not consider the step which is possible but does not help to solve the
equation.
\[
\sqrt{3}\sin x = 2 -\cos x
\]
\(3\sin ^{2}x = 4 - 4\cos x +\cos ^{2}x\)
substitution \( 2 -\cos x = z\)
\(3\sin ^{2}x = 4 -\cos ^{2}x\)
\(3\sin ^{2}x = 1 - 2\cos x +\cos ^{2}x\)