Trigonometric equations and inequalities

1003086009

Level:

The solution set of the equation

\sin x + \cos x = 0

for

x \in R

is:

⋃_{k \in Z} {\frac{3 π}{4} + 2 k π; \frac{7 π}{4} + 2 k π}

⋃_{k \in Z} {\frac{3 π}{4} + 2 k π}

R

\emptyset

1003086103

Level:

The solution set of the equation

2 \sin x + tg x = 0

x \in R

is:

⋃_{k \in Z} {k π; \frac{2 π}{3} + 2 k π; \frac{4 π}{3} + 2 k π}

⋃_{k \in Z} {2 k π; \frac{2 π}{3} + 2 k π; \frac{4 π}{3} + 2 k π}

⋃_{k \in Z} {k π; \frac{5 π}{6} + k π; \frac{7 π}{6} + k π}

⋃_{k \in Z} {k π; \frac{5 π}{6} + 2 k π; \frac{7 π}{6} + 2 k π}

1003086105

Level:

How many solutions does the equation

2 \cos^{2} x - \cos x - 1 = 0

have in the interval

[- π; 3 π]

6

4

2

3

1003086107

Level:

The solution set of the equation

2 {tg}^{2} x + 4 \cos^{2} x = 7

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} + 2 k π; \frac{2 π}{3} + 2 k π}

⋃_{k \in Z} {\frac{π}{3} + k π}

⋃_{k \in Z} {\frac{2 π}{3} + k π}

1003086108

Level:

The solution set of the equation

1 + \sin x \cdot \cos 2 x = \sin x + \cos 2 x

for

x \in R

is:

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

⋃_{k \in Z} {\frac{π}{2} + 2 k π}

⋃_{k \in Z} {k π}

⋃_{k \in Z} {2 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}

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]

9000046505

Level:

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:

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:

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

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Trigonometric equations and inequalities

1003086009

1003086103

1003086105

1003086107

1003086108

2010009804

2010009805

9000046505

9000046507

9000046508

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