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-1<(0.2)/(4*cos^2(x)-3)<0

Solution

- 1 < \frac{0.2}{4 \cdot cos ^{2} ( x ) - 3} < 0

Solution

False for all x \in R

Solution steps

- 1 < \frac{0.2}{4 cos ^{2} ( x ) - 3} < 0

a < u < b

then

a < u and u < b

- 1 < \frac{0.2}{4 cos ^{2} ( x ) - 3} and \frac{0.2}{4 cos ^{2} ( x ) - 3} < 0

- 1 < \frac{0.2}{4 cos ^{2} ( x ) - 3} : - \frac{π}{6} + 2 πn < x < \frac{π}{6} + 2 πn or arccos (0.83666 \dots) + 2 πn < x < arccos (- 0.83666 \dots) + 2 πn or \frac{5 π}{6} + 2 πn < x < \frac{7 π}{6} + 2 πn or 3.72123 \dots + 2 πn < x < 5.70354 \dots + 2 πn

- 1 < \frac{0.2}{4 cos ^{2} ( x ) - 3}

Switch sides

\frac{0.2}{4 cos ^{2} ( x ) - 3} > - 1

Rewrite in standard form

\frac{0.2}{4 cos ^{2} ( x ) - 3} > - 1

Add

1

to both sides

\frac{0.2}{4 cos ^{2} ( x ) - 3} + 1 > - 1 + 1

Simplify

\frac{0.2}{4 cos ^{2} ( x ) - 3} + 1 > 0

Simplify

\frac{0.2}{4 cos ^{2} ( x ) - 3} + 1 : \frac{4 cos ^{2} ( x ) - 2.8}{4 cos ^{2} ( x ) - 3}

\frac{0.2}{4 cos ^{2} ( x ) - 3} + 1

Convert element to fraction:

1 = \frac{1 ( 4 cos ^{2} ( x ) - 3 )}{4 cos ^{2} ( x ) - 3}

= \frac{0.2}{4 cos ^{2} ( x ) - 3} + \frac{1 \cdot ( 4 cos ^{2} ( x ) - 3 )}{4 cos ^{2} ( x ) - 3}

Since the denominators are equal, combine the fractions:

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

= \frac{0.2 + 1 \cdot ( 4 cos ^{2} ( x ) - 3 )}{4 cos ^{2} ( x ) - 3}

0.2 + 1 \cdot (4 cos^{2} (x) - 3) = 4 cos^{2} (x) - 2.8

0.2 + 1 \cdot (4 cos^{2} (x) - 3)

1 \cdot (4 cos^{2} (x) - 3) = 4 cos^{2} (x) - 3

1 \cdot (4 cos^{2} (x) - 3)

Multiply:

1 \cdot (4 cos^{2} (x) - 3) = (4 cos^{2} (x) - 3)

= 4 cos^{2} (x) - 3

Remove parentheses:

(a) = a

= 4 cos^{2} (x) - 3

= 0.2 + 4 cos^{2} (x) - 3

Group like terms

= 4 cos^{2} (x) + 0.2 - 3

Add/Subtract the numbers:

0.2 - 3 = - 2.8

= 4 cos^{2} (x) - 2.8

= \frac{4 cos ^{2} ( x ) - 2.8}{4 cos ^{2} ( x ) - 3}

\frac{4 cos ^{2} ( x ) - 2.8}{4 cos ^{2} ( x ) - 3} > 0

\frac{4 cos ^{2} ( x ) - 2.8}{4 cos ^{2} ( x ) - 3} > 0

Factor

\frac{4 cos ^{2} ( x ) - 2.8}{4 cos ^{2} ( x ) - 3} : \frac{( 2 cos ( x ) + 2.8 ) ( 2 cos ( x ) - 2.8 )}{( 2 cos ( x ) + 3 ) ( 2 cos ( x ) - 3 )}

\frac{4 cos ^{2} ( x ) - 2.8}{4 cos ^{2} ( x ) - 3}

Factor

4 cos^{2} (x) - 3 : (2 cos (x) + 3) (2 cos (x) - 3)

4 cos^{2} (x) - 3

Rewrite

4 cos^{2} (x) - 3

(2 cos (x))^{2} - (3)^{2}

4 cos^{2} (x) - 3

Rewrite

4

2^{2}

= 2^{2} cos^{2} (x) - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= 2^{2} cos^{2} (x) - (3)^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

2^{2} cos^{2} (x) = (2 cos (x))^{2}

= (2 cos (x))^{2} - (3)^{2}

= (2 cos (x))^{2} - (3)^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(2 cos (x))^{2} - (3)^{2} = (2 cos (x) + 3) (2 cos (x) - 3)

= (2 cos (x) + 3) (2 cos (x) - 3)

= \frac{4 cos ^{2} ( x ) - 2.8}{( 2 cos ( x ) + 3 ) ( 2 cos ( x ) - 3 )}

Factor

4 cos^{2} (x) - 2.8 : (2 cos (x) + 2.8) (2 cos (x) - 2.8)

4 cos^{2} (x) - 2.8

Rewrite

4 cos^{2} (x) - 2.8

(2 cos (x))^{2} - (2.8)^{2}

4 cos^{2} (x) - 2.8

Rewrite

4

2^{2}

= 2^{2} cos^{2} (x) - 2.8

Apply radical rule:

a = (a)^{2}

2.8 = (2.8)^{2}

= 2^{2} cos^{2} (x) - (2.8)^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

2^{2} cos^{2} (x) = (2 cos (x))^{2}

= (2 cos (x))^{2} - (2.8)^{2}

= (2 cos (x))^{2} - (2.8)^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(2 cos (x))^{2} - (2.8)^{2} = (2 cos (x) + 2.8) (2 cos (x) - 2.8)

= (2 cos (x) + 2.8) (2 cos (x) - 2.8)

= \frac{( 2 cos ( x ) + 2.8 ) ( 2 cos ( x ) - 2.8 )}{( 2 cos ( x ) + 3 ) ( 2 cos ( x ) - 3 )}

\frac{( 2 cos ( x ) + 2.8 ) ( 2 cos ( x ) - 2.8 )}{( 2 cos ( x ) + 3 ) ( 2 cos ( x ) - 3 )} > 0

Identify the intervals

Find the signs of the factors of

\frac{( 2 cos ( x ) + 2.8 ) ( 2 cos ( x ) - 2.8 )}{( 2 cos ( x ) + 3 ) ( 2 cos ( x ) - 3 )}

Find the signs of

2 cos (x) + 2.8

2 cos (x) + 2.8 = 0 : cos (x) = - 0.83666 \dots

2 cos (x) + 2.8 = 0

Move

2.8

to the right side

2 cos (x) + 2.8 = 0

Subtract

2.8

from both sides

2 cos (x) + 2.8 - 2.8 = 0 - 2.8

Simplify

2 cos (x) = - 1.67332 \dots

2 cos (x) = - 1.67332 \dots

Divide both sides by

2

2 cos (x) = - 1.67332 \dots

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{- 1.67332 \dots}{2}

Simplify

\frac{2 cos ( x )}{2} = \frac{- 1.67332 \dots}{2}

Simplify

\frac{2 cos ( x )}{2} : cos (x)

\frac{2 cos ( x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= cos (x)

Simplify

\frac{- 1.67332 \dots}{2} : - 0.83666 \dots

\frac{- 1.67332 \dots}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1.67332 \dots}{2}

Divide the numbers:

\frac{1.67332 \dots}{2} = 0.83666 \dots

= - 0.83666 \dots

cos (x) = - 0.83666 \dots

cos (x) = - 0.83666 \dots

cos (x) = - 0.83666 \dots

2 cos (x) + 2.8 < 0 : cos (x) < - 0.83666 \dots

2 cos (x) + 2.8 < 0

Move

2.8

to the right side

2 cos (x) + 2.8 < 0

Subtract

2.8

from both sides

2 cos (x) + 2.8 - 2.8 < 0 - 2.8

Simplify

2 cos (x) < - 1.67332 \dots

2 cos (x) < - 1.67332 \dots

Divide both sides by

2

2 cos (x) < - 1.67332 \dots

Divide both sides by

2

\frac{2 cos ( x )}{2} < \frac{- 1.67332 \dots}{2}

Simplify

\frac{2 cos ( x )}{2} < \frac{- 1.67332 \dots}{2}

Simplify

\frac{2 cos ( x )}{2} : cos (x)

\frac{2 cos ( x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= cos (x)

Simplify

\frac{- 1.67332 \dots}{2} : - 0.83666 \dots

\frac{- 1.67332 \dots}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1.67332 \dots}{2}

Divide the numbers:

\frac{1.67332 \dots}{2} = 0.83666 \dots

= - 0.83666 \dots

cos (x) < - 0.83666 \dots

cos (x) < - 0.83666 \dots

cos (x) < - 0.83666 \dots

2 cos (x) + 2.8 > 0 : cos (x) > - 0.83666 \dots

2 cos (x) + 2.8 > 0

Move

2.8

to the right side

2 cos (x) + 2.8 > 0

Subtract

2.8

from both sides

2 cos (x) + 2.8 - 2.8 > 0 - 2.8

Simplify

2 cos (x) > - 1.67332 \dots

2 cos (x) > - 1.67332 \dots

Divide both sides by

2

2 cos (x) > - 1.67332 \dots

Divide both sides by

2

\frac{2 cos ( x )}{2} > \frac{- 1.67332 \dots}{2}

Simplify

\frac{2 cos ( x )}{2} > \frac{- 1.67332 \dots}{2}

Simplify

\frac{2 cos ( x )}{2} : cos (x)

\frac{2 cos ( x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= cos (x)

Simplify

\frac{- 1.67332 \dots}{2} : - 0.83666 \dots

\frac{- 1.67332 \dots}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1.67332 \dots}{2}

Divide the numbers:

\frac{1.67332 \dots}{2} = 0.83666 \dots

= - 0.83666 \dots

cos (x) > - 0.83666 \dots

cos (x) > - 0.83666 \dots

cos (x) > - 0.83666 \dots

Find the signs of

2 cos (x) - 2.8

2 cos (x) - 2.8 = 0 : cos (x) = 0.83666 \dots

2 cos (x) - 2.8 = 0

Move

2.8

to the right side

2 cos (x) - 2.8 = 0

Add

2.8

to both sides

2 cos (x) - 2.8 + 2.8 = 0 + 2.8

Simplify

2 cos (x) = 1.67332 \dots

2 cos (x) = 1.67332 \dots

Divide both sides by

2

2 cos (x) = 1.67332 \dots

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{1.67332 \dots}{2}

Simplify

cos (x) = 0.83666 \dots

cos (x) = 0.83666 \dots

2 cos (x) - 2.8 < 0 : cos (x) < 0.83666 \dots

2 cos (x) - 2.8 < 0

Move

2.8

to the right side

2 cos (x) - 2.8 < 0

Add

2.8

to both sides

2 cos (x) - 2.8 + 2.8 < 0 + 2.8

Simplify

2 cos (x) < 1.67332 \dots

2 cos (x) < 1.67332 \dots

Divide both sides by

2

2 cos (x) < 1.67332 \dots

Divide both sides by

2

\frac{2 cos ( x )}{2} < \frac{1.67332 \dots}{2}

Simplify

cos (x) < 0.83666 \dots

cos (x) < 0.83666 \dots

2 cos (x) - 2.8 > 0 : cos (x) > 0.83666 \dots

2 cos (x) - 2.8 > 0

Move

2.8

to the right side

2 cos (x) - 2.8 > 0

Add

2.8

to both sides

2 cos (x) - 2.8 + 2.8 > 0 + 2.8

Simplify

2 cos (x) > 1.67332 \dots

2 cos (x) > 1.67332 \dots

Divide both sides by

2

2 cos (x) > 1.67332 \dots

Divide both sides by

2

\frac{2 cos ( x )}{2} > \frac{1.67332 \dots}{2}

Simplify

cos (x) > 0.83666 \dots

cos (x) > 0.83666 \dots

Find the signs of

2 cos (x) + 3

2 cos (x) + 3 = 0 : cos (x) = - \frac{3}{2}

2 cos (x) + 3 = 0

Move

3

to the right side

2 cos (x) + 3 = 0

Subtract

3

from both sides

2 cos (x) + 3 - 3 = 0 - 3

Simplify

2 cos (x) = - 3

2 cos (x) = - 3

Divide both sides by

2

2 cos (x) = - 3

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{- 3}{2}

Simplify

cos (x) = - \frac{3}{2}

cos (x) = - \frac{3}{2}

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

2 cos (x) + 3 < 0

Move

3

to the right side

2 cos (x) + 3 < 0

Subtract

3

from both sides

2 cos (x) + 3 - 3 < 0 - 3

Simplify

2 cos (x) < - 3

2 cos (x) < - 3

Divide both sides by

2

2 cos (x) < - 3

Divide both sides by

2

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

Simplify

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

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

2 cos (x) + 3 > 0 : cos (x) > - \frac{3}{2}

2 cos (x) + 3 > 0

Move

3

to the right side

2 cos (x) + 3 > 0

Subtract

3

from both sides

2 cos (x) + 3 - 3 > 0 - 3

Simplify

2 cos (x) > - 3

2 cos (x) > - 3

Divide both sides by

2

2 cos (x) > - 3

Divide both sides by

2

\frac{2 cos ( x )}{2} > \frac{- 3}{2}

Simplify

cos (x) > - \frac{3}{2}

cos (x) > - \frac{3}{2}

Find the signs of

2 cos (x) - 3

2 cos (x) - 3 = 0 : cos (x) = \frac{3}{2}

2 cos (x) - 3 = 0

Move

3

to the right side

2 cos (x) - 3 = 0

Add

3

to both sides

2 cos (x) - 3 + 3 = 0 + 3

Simplify

2 cos (x) = 3

2 cos (x) = 3

Divide both sides by

2

2 cos (x) = 3

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{3}{2}

Simplify

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2}

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

2 cos (x) - 3 < 0

Move

3

to the right side

2 cos (x) - 3 < 0

Add

3

to both sides

2 cos (x) - 3 + 3 < 0 + 3

Simplify

2 cos (x) < 3

2 cos (x) < 3

Divide both sides by

2

2 cos (x) < 3

Divide both sides by

2

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

Simplify

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

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

2 cos (x) - 3 > 0 : cos (x) > \frac{3}{2}

2 cos (x) - 3 > 0

Move

3

to the right side

2 cos (x) - 3 > 0

Add

3

to both sides

2 cos (x) - 3 + 3 > 0 + 3

Simplify

2 cos (x) > 3

2 cos (x) > 3

Divide both sides by

2

2 cos (x) > 3

Divide both sides by

2

\frac{2 cos ( x )}{2} > \frac{3}{2}

Simplify

cos (x) > \frac{3}{2}

cos (x) > \frac{3}{2}

Find singularity points

Find the zeros of the denominator

(2 cos (x) + 3) (2 cos (x) - 3) :

No Solution

(2 cos (x) + 3) (2 cos (x) - 3) = 0

The sides are not equal

No Solution

Summarize in a table:

2 cos (x) + 2.8 2 cos (x) - 2.8 2 cos (x) + 3 2 cos (x) - 3 \frac{( 2 c o s ( x ) + 2.8 ) ( 2 c o s ( x ) - 2.8 )}{( 2 c o s ( x ) + 3 ) ( 2 c o s ( x ) - 3 )} cos (x) < - \frac{3}{2} - - - - + cos (x) = - \frac{3}{2} - - 0 - Undefined - \frac{3}{2} < cos (x) < - 0.83666 \dots - - + - - cos (x) = - 0.83666 \dots 0 - + - 0 - 0.83666 \dots < cos (x) < 0.83666 \dots + - + - + cos (x) = 0.83666 \dots + 0 + - 0 0.83666 \dots < cos (x) < \frac{3}{2} + + + - - cos (x) = \frac{3}{2} + + + 0 Undefined cos (x) > \frac{3}{2} + + + + +

Identify the intervals that satisfy the required condition:

> 0

cos (x) < - \frac{3}{2} or - 0.83666 \dots < cos (x) < 0.83666 \dots or cos (x) > \frac{3}{2}

cos (x) < - \frac{3}{2} or - 0.83666 \dots < cos (x) < 0.83666 \dots or cos (x) > \frac{3}{2}

cos (x) < - \frac{3}{2} : \frac{5 π}{6} + 2 πn < x < \frac{7 π}{6} + 2 πn

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

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

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

Simplify

arccos (- \frac{3}{2}) : \frac{5 π}{6}

arccos (- \frac{3}{2})

Use the following trivial identity:

arccos (- \frac{3}{2}) = \frac{5 π}{6}

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{5 π}{6}

Simplify

2 π - arccos (- \frac{3}{2}) : \frac{7 π}{6}

2 π - arccos (- \frac{3}{2})

Use the following trivial identity:

arccos (- \frac{3}{2}) = \frac{5 π}{6}

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{5 π}{6}

Simplify

2 π - \frac{5 π}{6}

Convert element to fraction:

2 π = \frac{2 π 6}{6}

= \frac{2 π 6}{6} - \frac{5 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π 6 - 5 π}{6}

2 π 6 - 5 π = 7 π

2 π 6 - 5 π

Multiply the numbers:

2 \cdot 6 = 12

= 12 π - 5 π

Add similar elements:

12 π - 5 π = 7 π

= 7 π

= \frac{7 π}{6}

= \frac{7 π}{6}

\frac{5 π}{6} + 2 πn < x < \frac{7 π}{6} + 2 πn

- 0.83666 \dots < cos (x) < 0.83666 \dots : arccos (0.83666 \dots) + 2 πn < x < arccos (- 0.83666 \dots) + 2 πn or 3.72123 \dots + 2 πn < x < 5.70354 \dots + 2 πn

- 0.83666 \dots < cos (x) < 0.83666 \dots

a < u < b

then

a < u and u < b

- 0.83666 \dots < cos (x) and cos (x) < 0.83666 \dots

- 0.83666 \dots < cos (x) : - 2.56195 \dots + 2 πn < x < arccos (- 0.83666 \dots) + 2 πn

- 0.83666 \dots < cos (x)

Switch sides

cos (x) > - 0.83666 \dots

For

cos (x) > a

, if

- 1 \leq a < 1

then

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

- arccos (- 0.83666 \dots) + 2 πn < x < arccos (- 0.83666 \dots) + 2 πn

Simplify

- 2.56195 \dots + 2 πn < x < arccos (- 0.83666 \dots) + 2 πn

cos (x) < 0.83666 \dots : arccos (0.83666 \dots) + 2 πn < x < 5.70354 \dots + 2 πn

cos (x) < 0.83666 \dots

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

arccos (0.83666 \dots) + 2 πn < x < 2 π - arccos (0.83666 \dots) + 2 πn

Simplify

arccos (0.83666 \dots) + 2 πn < x < 5.70354 \dots + 2 πn

Combine the intervals

- 2.56195 \dots + 2 πn < x < arccos (- 0.83666 \dots) + 2 πn and arccos (0.83666 \dots) + 2 πn < x < 5.70354 \dots + 2 πn

Merge Overlapping Intervals

arccos (0.83666 \dots) + 2 πn < x < arccos (- 0.83666 \dots) + 2 πn or 3.72123 \dots + 2 πn < x < 5.70354 \dots + 2 πn

cos (x) > \frac{3}{2} : - \frac{π}{6} + 2 πn < x < \frac{π}{6} + 2 πn

cos (x) > \frac{3}{2}

For

cos (x) > a

, if

- 1 \leq a < 1

then

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

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

Simplify

- arccos (\frac{3}{2}) : - \frac{π}{6}

- arccos (\frac{3}{2})

Use the following trivial identity:

arccos (\frac{3}{2}) = \frac{π}{6}

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}{6}

Simplify

arccos (\frac{3}{2}) : \frac{π}{6}

arccos (\frac{3}{2})

Use the following trivial identity:

arccos (\frac{3}{2}) = \frac{π}{6}

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}{6}

- \frac{π}{6} + 2 πn < x < \frac{π}{6} + 2 πn

Combine the intervals

\frac{5 π}{6} + 2 πn < x < \frac{7 π}{6} + 2 πn or (arccos (0.83666 \dots) + 2 πn < x < arccos (- 0.83666 \dots) + 2 πn or 3.72123 \dots + 2 πn < x < 5.70354 \dots + 2 πn) or - \frac{π}{6} + 2 πn < x < \frac{π}{6} + 2 πn

Merge Overlapping Intervals

- \frac{π}{6} + 2 πn < x < \frac{π}{6} + 2 πn or arccos (0.83666 \dots) + 2 πn < x < arccos (- 0.83666 \dots) + 2 πn or \frac{5 π}{6} + 2 πn < x < \frac{7 π}{6} + 2 πn or 3.72123 \dots + 2 πn < x < 5.70354 \dots + 2 πn

\frac{0.2}{4 cos ^{2} ( x ) - 3} < 0 :

No Solution for

x \in R

\frac{0.2}{4 cos ^{2} ( x ) - 3} < 0

Divide both sides by

0.2

\frac{\frac{0.2}{4 c o s ^{2} ( x ) - 3}}{0.2} < \frac{0}{0.2}

Simplify

\frac{1}{4 cos ^{2} ( x ) - 3} < 0

\frac{1}{a} < 0

then

a < 0

4 cos^{2} (x) - 3 < 0

No Solution for x \in R

Combine the intervals

(- \frac{π}{6} + 2 πn < x < \frac{π}{6} + 2 πn or arccos (0.83666 \dots) + 2 πn < x < arccos (- 0.83666 \dots) + 2 πn or \frac{5 π}{6} + 2 πn < x < \frac{7 π}{6} + 2 πn or 3.72123 \dots + 2 πn < x < 5.70354 \dots + 2 πn) and False for all x \in R

Merge Overlapping Intervals

False for all x \in R

Popular Examples

-180<tan(x)<180

- 18 0^{\circ} < tan (x) < 18 0^{\circ}

-pi/2 <arctan(y)< pi/2

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sin(θ)<0\land sec(θ)>0

sin (θ) < 0 and sec (θ) > 0

cos(θ)= 3/4 \land cot(θ)<0

cos (θ) = \frac{3}{4} and cot (θ) < 0

sin(θ)>0\land cos(θ)>0

sin (θ) > 0 and cos (θ) > 0