English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos((3x)/2)<0

Solution

cos (\frac{3 x}{2}) < 0

Solution

\frac{π}{3} + \frac{4 π}{3} n < x < π + \frac{4 π}{3} n

Interval Notation

(\frac{π}{3} + \frac{4 π}{3} n, π + \frac{4 π}{3} n)

Decimal Notation

1.04719 \dots + \frac{4 π}{3} n < x < 3.14159 \dots + \frac{4 π}{3} n

Solution steps

cos (\frac{3 x}{2}) < 0

For

cos (x) < a

, if

- 1 < a \leq 1

then

arccos (a) + 2 πn < x < 2 π - arccos (a) + 2 πn

arccos (0) + 2 πn < \frac{3 x}{2} < 2 π - arccos (0) + 2 πn

a < u < b

then

a < u and u < b

arccos (0) + 2 πn < \frac{3 x}{2} and \frac{3 x}{2} < 2 π - arccos (0) + 2 πn

arccos (0) + 2 πn < \frac{3 x}{2} : x > \frac{π}{3} + \frac{4 πn}{3}

arccos (0) + 2 πn < \frac{3 x}{2}

Switch sides

\frac{3 x}{2} > arccos (0) + 2 πn

Simplify

arccos (0) + 2 πn : \frac{π}{2} + 2 πn

arccos (0) + 2 πn

Use the following trivial identity:

arccos (0) = \frac{π}{2}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{2} + 2 πn

\frac{3 x}{2} > \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{3 x}{2} > \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} > 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 3 x}{2} > 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

3 x > π + 4 πn

3 x > π + 4 πn

3 x > π + 4 πn

Divide both sides by

3

3 x > π + 4 πn

Divide both sides by

3

\frac{3 x}{3} > \frac{π}{3} + \frac{4 πn}{3}

Simplify

x > \frac{π}{3} + \frac{4 πn}{3}

x > \frac{π}{3} + \frac{4 πn}{3}

\frac{3 x}{2} < 2 π - arccos (0) + 2 πn : x < π + \frac{4 πn}{3}

\frac{3 x}{2} < 2 π - arccos (0) + 2 πn

Simplify

2 π - arccos (0) + 2 πn : 2 π - \frac{π}{2} + 2 πn

2 π - arccos (0) + 2 πn

Use the following trivial identity:

arccos (0) = \frac{π}{2}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{π}{2} + 2 πn

\frac{3 x}{2} < 2 π - \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{3 x}{2} < 2 π - \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 \cdot 3 x}{2} < 2 \cdot 2 π - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 3 x}{2} < 2 \cdot 2 π - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

2 \cdot 2 π - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot 2 π - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot 2 π = 4 π

2 \cdot 2 π

Multiply the numbers:

2 \cdot 2 = 4

= 4 π

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 4 π - π + 4 πn

Add similar elements:

4 π - π = 3 π

= 3 π + 4 πn

3 x < 3 π + 4 πn

3 x < 3 π + 4 πn

3 x < 3 π + 4 πn

Divide both sides by

3

3 x < 3 π + 4 πn

Divide both sides by

3

\frac{3 x}{3} < \frac{3 π}{3} + \frac{4 πn}{3}

Simplify

x < π + \frac{4 πn}{3}

x < π + \frac{4 πn}{3}

Combine the intervals

x > \frac{π}{3} + \frac{4 πn}{3} and x < π + \frac{4 πn}{3}

Merge Overlapping Intervals

\frac{π}{3} + \frac{4 π}{3} n < x < π + \frac{4 π}{3} n

Popular Examples

sec(x)<0,csc(x)>0,0<= x<= 2pi

sec (x) < 0, csc (x) > 0, 0 \leq x \leq 2 π

2(cos(3x))^2+sqrt(3)sin(6x)<1

2 (cos (3 x))^{2} + 3 sin (6 x) < 1

2cos^2(x)-cos(x)-1<0

2 cos^{2} (x) - cos (x) - 1 < 0

sin(x/2)>0

sin (\frac{x}{2}) > 0

solvefor x,tan(x)<= 1 solve for

x, tan (x) \leq 1