Trigonometric Equations and Inequalities

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 \(\left (\frac{5\pi } {6}, \frac{3\pi } {2}\right )\).
\(\cos x < \frac{1} {2}\)
\(\mathop{\mathrm{tg}}\nolimits x < 0\)
\(\sin x\geq -\frac{\sqrt{2}} {2} \)
\(\mathop{\mathrm{cotg}}\nolimits x < 1\)

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 \( 2 -\cos x = z\)
\(3\sin ^{2}x = 4 -\cos ^{2}x\)
\(3\sin ^{2}x = 1 - 2\cos x +\cos ^{2}x\)

9000046510

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. \[ 2\sin ^{2}x -\sin x - 1 = 0 \]
substitution \( \sin x = z\)
substitution \( \sin ^{2}x = z\)
\(2\sin ^{2}x -\sin x = 1\)
\(2\sin ^{2}x -\sin x =\sin ^{2}x +\cos ^{2}x\)

9000046506

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. \[ \sin 2x =\mathop{\mathrm{tg}}\nolimits x \]
\(2\sin x\cdot \cos x = \frac{\sin x} {\cos x}\)
substitution \( 2x = z\)
\(\sin x = \frac{\mathop{\mathrm{tg}}\nolimits x} {2} \)
\(\cos ^{2}x -\sin ^{2}x =\mathop{\mathrm{tg}}\nolimits x\)

9000046509

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. \[ 2\cos ^{2}x =\sin x + 1 \]
\(2 - 2\sin ^{2}x =\sin x + 1\)
substitution \( \sin x + 1 = z\)
substitution \( \cos x = z\)
\(2\cos ^{2}x = \sqrt{1 -\sin ^{2 } x} + 1\)

9000046502

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. \[ \cos 3x = 0.5 \]
substitution \( 3x = z\)
substitution \( \cos x = z\)
\(\cos ^{3}x -\sin ^{3}x = 0.5\)
\(\cos x = \frac{0.5} {3} \)