Trigonometric equations and inequalities

2010010703

Level:

Identify the equation which arises from the following equation using an optimal substitution.

2 \sin^{2} x - 5 \cos x + 1 = 0

2 t^{2} + 5 t - 3 = 0

2 t^{2} - 5 t + 1 = 0

2 t^{2} + 5 t - 4 = 0

2 t^{2} - 5 t + 2 = 0

2010010702

Level:

The solution set of the equation

cotg x = \sqrt{3}

for

x \in (- π; π)

is:

{- \frac{5 π}{6}; \frac{π}{6}}

{- \frac{π}{6}; \frac{π}{6}}

{- \frac{π}{3}; \frac{π}{3}}

{- \frac{2 π}{3}; \frac{π}{3}}

2010010701

Level:

The solution set of the equation

\cos x = - 0.5

for

x \in [0; 2 π]

is:

{\frac{2 π}{3}; \frac{4 π}{3}}

{\frac{2 π}{3}; \frac{5 π}{3}}

{\frac{4 π}{3}; \frac{5 π}{3}}

{\frac{4 π}{3}; \frac{7 π}{3}}

2010009805

Level:

The solution set of the inequality

| \cos x | \leq \frac{1}{2}

for

x \in R

is:

⋃_{k \in Z} [\frac{π}{3} + k π; \frac{2 π}{3} + k π]

⋃_{k \in Z} [- \frac{π}{3} + k π; \frac{π}{3} + k π]

⋃_{k \in Z} [\frac{π}{3} + k π; \infty)

⋃_{k \in Z} [\frac{π}{3} + k π; \frac{4 π}{3} + k π]

2010009804

Level:

The solution set of the equation

tg x - cotg x = 0

for

x \in R

is:

⋃_{k \in Z} {\frac{π}{4} + k π; \frac{3 π}{4} + k π}

⋃_{k \in Z} {k π; \frac{π}{4} + k π}

⋃_{k \in Z} {\frac{π}{4} + k π}

⋃_{k \in Z} {\frac{3 π}{4} + k π}

2010009803

Level:

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

[- \frac{π}{2}; \frac{π}{2}]

3 \cos x - 2 = 0

3 \sin x - 2 = 0

2 \cos x - 3 = 0

3 \cos x + 2 = 0

2010009802

Level:

How many solutions does the equation

{cotg}^{2} x = 3

have for

- π \leq x \leq π

4

solutions

2

solutions

8

solutions

6

solutions

2010009801

Level:

How many solutions does the equation

\sin^{2} x = 0.75

have for

0 \leq x \leq 2 π

4

solutions

1

solution

2

solutions

3

solutions

2000006604

Level:

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

cotg x \geq - \frac{\sqrt{3}}{3}

x \in (- π; π) ∖ {0}

cotg x \geq \frac{1}{2}

x \in (- π; π) ∖ {0}

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

x \in (- π; π) ∖ {0}

cotg x \leq \frac{\sqrt{3}}{3}

x \in (- π; π) ∖ {0}

2000006603

Level:

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

cotg x \leq 1

x \in (- π; π) ∖ {0}

cotg x \geq 1

x \in (- π; π) ∖ {0}

tg x \leq 1

x \in (- π; π) ∖ {0}

tg x \geq 1

x \in (- π; π) ∖ {0}

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Trigonometric equations and inequalities

2010010703

2010010702

2010010701

2010009805

2010009804

2010009803

2010009802

2010009801

2000006604

2000006603

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