Trigonometric equations and inequalities

2010010705

Level:

The solution set of the equation

\cos (2 φ + \frac{π}{6}) = - 1

for

φ \in [0; 2 π]

is:

{\frac{5 π}{12}; \frac{17 π}{12}}

{\frac{5 π}{12}; \frac{11 π}{12}}

{\frac{7 π}{12}; \frac{13 π}{12}}

{\frac{7 π}{12}; \frac{17 π}{12}}

2010012001

Level:

Find all

x

x \in R

, such that

{tg}^{2} x = tg x

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

x \in ⋃_{k \in Z} {k π}

x \in ⋃_{k \in Z} {\frac{π}{4} + k π}

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

2010012002

Level:

Solve

\cos^{2} x = \sqrt{2} \cos x

for

x

, where

x \in R

x \in ⋃_{k \in Z} {\frac{π}{2} + k π}

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

x \in ⋃_{k \in Z} {\frac{π}{4} + k π}

x \in \emptyset

2010012003

Level:

The largest value of

φ

, where

0^{\circ} < φ < 360^{\circ}

, satisfying the equation

\sin (3 φ + 66^{\circ}) = - \frac{1}{2}

is:

328^{\circ}

208^{\circ}

338^{\circ}

288^{\circ}

2010012004

Level:

The smallest value of

φ

, where

0^{\circ} < φ < 180^{\circ}

, satisfying the equation

tg (2 φ + 34^{\circ}) = - \frac{\sqrt{3}}{3}

is:

58^{\circ}

148^{\circ}

29^{\circ}

92^{\circ}

2010012005

Level:

The arithmetic mean of all values of

φ

between

0^{\circ}

and

360^{\circ}

for which

\sin (φ - 40^{\circ}) = 0

is:

130^{\circ}

220^{\circ}

40^{\circ}

180^{\circ}

9000046502

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.

\cos 3 x = 0.5

substitution

3 x = z

substitution

\cos x = z

\cos^{3} x - \sin^{3} x = 0.5

\cos x = \frac{0.5}{3}

9000046503

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.

tg (- x + \frac{π}{6}) = \sqrt{3}

substitution

- x + \frac{π}{6} = z

tg (- x) = \sqrt{3} - \frac{π}{6}

{tg}^{2} (- x + \frac{π}{6}) = 3

\frac{\sin (- x + \frac{π}{6})}{\cos (- x + \frac{π}{6})} = \sqrt{3}

9000046504

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.

\cos (x + \frac{π}{3}) = \frac{\sqrt{3}}{2}

substitution

x + \frac{π}{3} = z

\cos^{2} (x + \frac{π}{3}) = \frac{3}{4}

substitution

\frac{\sqrt{3}}{2} = z

\cos x \cdot \cos \frac{π}{3} - \sin x \cdot \sin \frac{π}{3} = \frac{\sqrt{3}}{2}

9000046510

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.

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

Trigonometric equations and inequalities

2010010705

2010012001

2010012002

2010012003

2010012004

2010012005

9000046502

9000046503

9000046504

9000046510

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