Trigonometric equations and inequalities

1003085903

Level:

The solution set of the inequality

\sin^{2} x \leq 1

for

x \in R

is:

R

\emptyset

R ∖ {\frac{π}{2} + 2 k π : k \in Z}

R ∖ {\frac{3 π}{2} + 2 k π : k \in Z}

1003085904

Level:

The solution set of the inequality

\cos^{2} x < - 1

for

x \in R

is:

\emptyset

R

⋃_{k \in Z} (k π; 2 k π)

⋃_{k \in Z} (\frac{3 k π}{2}; 2 k π)

1003086008

Level:

The solution set of the equation

tg x \cdot cotg x = 1

for

x \in R

is:

R ∖ {\frac{k π}{2} : k \in Z}

R

R ∖ {k π : k \in Z}

R ∖ {2 k π : k \in Z}

1003086010

Level:

The sum of all

x \in [0; 3 π]

satisfying the equation

\frac{tg 0.5 x}{3} = \frac{1}{\sqrt{27}}

is:

\frac{8 π}{3}

\frac{π}{6}

\frac{4 π}{3}

\frac{7 π}{3}

1003086101

Level:

Which of the following equations has exactly four solutions in the interval

[0; π]

tg 4 x = 2

tg 2 x = 4

tg \frac{x}{4} = 2

tg \frac{x}{4} = 4

1003086102

Level:

How many solutions does the equation

cotg (2 x - \frac{π}{4}) = \frac{1}{\sqrt{3}}

have in the interval

[0; 2 π]

4

8

2

0

1003086106

Level:

The solution set of the equation

\sin 2 x = \cos 3 x \cdot \sin 2 x

for

x \in [0^{\circ}; 180^{\circ}]

is:

{0^{\circ}; 90^{\circ}; 120^{\circ}; 180^{\circ}}

{90^{\circ}; 120^{\circ}; 180^{\circ}}

{90^{\circ}; 180^{\circ}}

{0^{\circ}; 90^{\circ}}

1003086109

Level:

The solution set of the equation

tg x + cotg x = 2

for

x \in [- 2 π; 2 π]

is:

{- \frac{7 π}{4}; - \frac{3 π}{4}; \frac{π}{4}; \frac{5 π}{4}}

{- \frac{7 π}{4}; \frac{π}{4};}

{- \frac{5 π}{4}; - \frac{π}{4}; \frac{3 π}{4}; \frac{7 π}{4}}

{- \frac{3 π}{4}; \frac{π}{4}}

1003086110

Level:

Which of the following intervals contains four solutions of the equation

\cos x = 0.5

[π; 5 π]

[- 2 π; π]

[- 4 π; - 2 π]

[5 π; 10 π]

2000006301

Level:

An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.

\sin x \geq \frac{\sqrt{3}}{2}

x \in [0; 2 π]

\sin x \geq \frac{\sqrt{2}}{2}

x \in [0; 2 π]

\sin x > \frac{\sqrt{3}}{2}

x \in [0; 2 π]

\sin x > \frac{\sqrt{2}}{2}

x \in [0; 2 π]

Trigonometric equations and inequalities

1003085903

1003085904

1003086008

1003086010

1003086101

1003086102

1003086106

1003086109

1003086110

2000006301

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