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(sin(x))/(4cos^2(x)-1)<0

Solution

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

Solution

\frac{π}{3} + 2 πn < x < \frac{2 π}{3} + 2 πn or π + 2 πn < x < \frac{4 π}{3} + 2 πn or \frac{5 π}{3} + 2 πn < x < 2 π + 2 πn

Interval Notation

(\frac{π}{3} + 2 πn, \frac{2 π}{3} + 2 πn) \cup (π + 2 πn, \frac{4 π}{3} + 2 πn) \cup (\frac{5 π}{3} + 2 πn, 2 π + 2 πn)

Decimal Notation

1.04719 \dots + 2 πn < x < 2.09439 \dots + 2 πn or 3.14159 \dots + 2 πn < x < 4.18879 \dots + 2 πn or 5.23598 \dots + 2 πn < x < 6.28318 \dots + 2 πn

Solution steps

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

Use the following identity:

cos^{2} (x) + sin^{2} (x) = 1

Therefore

cos^{2} (x) = 1 - sin^{2} (x)

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

Simplify

\frac{sin ( x )}{4 ( 1 - sin ^{2} ( x ) ) - 1} : \frac{sin ( x )}{- 4 sin ^{2} ( x ) + 3}

\frac{sin ( x )}{4 ( 1 - sin ^{2} ( x ) ) - 1}

Expand

4 (1 - sin^{2} (x)) - 1 : - 4 sin^{2} (x) + 3

4 (1 - sin^{2} (x)) - 1

Expand

4 (1 - sin^{2} (x)) : 4 - 4 sin^{2} (x)

4 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (x)

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

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 sin^{2} (x)

= 4 - 4 sin^{2} (x) - 1

Simplify

4 - 4 sin^{2} (x) - 1 : - 4 sin^{2} (x) + 3

4 - 4 sin^{2} (x) - 1

Group like terms

= - 4 sin^{2} (x) + 4 - 1

Add/Subtract the numbers:

4 - 1 = 3

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

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

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

\frac{sin ( x )}{- 4 sin ^{2} ( x ) + 3} < 0

Let:

u = sin (x)

\frac{u}{- 4 u ^{2} + 3} < 0

\frac{u}{- 4 u ^{2} + 3} < 0 : - \frac{3}{2} < u < 0 or u > \frac{3}{2}

\frac{u}{- 4 u ^{2} + 3} < 0

Factor

\frac{u}{- 4 u ^{2} + 3} : \frac{u}{- ( 2 u + 3 ) ( 2 u - 3 )}

\frac{u}{- 4 u ^{2} + 3}

Factor

- 4 u^{2} + 3 : - (2 u + 3) (2 u - 3)

- 4 u^{2} + 3

Factor out common term

- 1

= - (4 u^{2} - 3)

Factor

4 u^{2} - 3 : (2 u + 3) (2 u - 3)

4 u^{2} - 3

Rewrite

4 u^{2} - 3

(2 u)^{2} - (3)^{2}

4 u^{2} - 3

Rewrite

4

2^{2}

= 2^{2} u^{2} - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= 2^{2} u^{2} - (3)^{2}

Apply exponent rule:

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

2^{2} u^{2} = (2 u)^{2}

= (2 u)^{2} - (3)^{2}

= (2 u)^{2} - (3)^{2}

Apply Difference of Two Squares Formula:

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

(2 u)^{2} - (3)^{2} = (2 u + 3) (2 u - 3)

= (2 u + 3) (2 u - 3)

= - (2 u + 3) (2 u - 3)

= \frac{u}{- ( 2 u + 3 ) ( 2 u - 3 )}

\frac{u}{- ( 2 u + 3 ) ( 2 u - 3 )} < 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{u ( - 1 )}{- ( 2 u + 3 ) ( 2 u - 3 )} > 0 \cdot (- 1)

Simplify

\frac{u}{( 2 u + 3 ) ( 2 u - 3 )} > 0

Identify the intervals

Find the signs of the factors of

\frac{u}{( 2 u + 3 ) ( 2 u - 3 )}

Find the signs of

u

u = 0

u < 0

u > 0

Find the signs of

2 u + 3

2 u + 3 = 0 : u = - \frac{3}{2}

2 u + 3 = 0

Move

3

to the right side

2 u + 3 = 0

Subtract

3

from both sides

2 u + 3 - 3 = 0 - 3

Simplify

2 u = - 3

2 u = - 3

Divide both sides by

2

2 u = - 3

Divide both sides by

2

\frac{2 u}{2} = \frac{- 3}{2}

Simplify

u = - \frac{3}{2}

u = - \frac{3}{2}

2 u + 3 < 0 : u < - \frac{3}{2}

2 u + 3 < 0

Move

3

to the right side

2 u + 3 < 0

Subtract

3

from both sides

2 u + 3 - 3 < 0 - 3

Simplify

2 u < - 3

2 u < - 3

Divide both sides by

2

2 u < - 3

Divide both sides by

2

\frac{2 u}{2} < \frac{- 3}{2}

Simplify

u < - \frac{3}{2}

u < - \frac{3}{2}

2 u + 3 > 0 : u > - \frac{3}{2}

2 u + 3 > 0

Move

3

to the right side

2 u + 3 > 0

Subtract

3

from both sides

2 u + 3 - 3 > 0 - 3

Simplify

2 u > - 3

2 u > - 3

Divide both sides by

2

2 u > - 3

Divide both sides by

2

\frac{2 u}{2} > \frac{- 3}{2}

Simplify

u > - \frac{3}{2}

u > - \frac{3}{2}

Find the signs of

2 u - 3

2 u - 3 = 0 : u = \frac{3}{2}

2 u - 3 = 0

Move

3

to the right side

2 u - 3 = 0

Add

3

to both sides

2 u - 3 + 3 = 0 + 3

Simplify

2 u = 3

2 u = 3

Divide both sides by

2

2 u = 3

Divide both sides by

2

\frac{2 u}{2} = \frac{3}{2}

Simplify

u = \frac{3}{2}

u = \frac{3}{2}

2 u - 3 < 0 : u < \frac{3}{2}

2 u - 3 < 0

Move

3

to the right side

2 u - 3 < 0

Add

3

to both sides

2 u - 3 + 3 < 0 + 3

Simplify

2 u < 3

2 u < 3

Divide both sides by

2

2 u < 3

Divide both sides by

2

\frac{2 u}{2} < \frac{3}{2}

Simplify

u < \frac{3}{2}

u < \frac{3}{2}

2 u - 3 > 0 : u > \frac{3}{2}

2 u - 3 > 0

Move

3

to the right side

2 u - 3 > 0

Add

3

to both sides

2 u - 3 + 3 > 0 + 3

Simplify

2 u > 3

2 u > 3

Divide both sides by

2

2 u > 3

Divide both sides by

2

\frac{2 u}{2} > \frac{3}{2}

Simplify

u > \frac{3}{2}

u > \frac{3}{2}

Find singularity points

Find the zeros of the denominator

(2 u + 3) (2 u - 3) : u = - \frac{3}{2}, u = \frac{3}{2}

(2 u + 3) (2 u - 3) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

2 u + 3 = 0 or 2 u - 3 = 0

Solve

2 u + 3 = 0 : u = - \frac{3}{2}

2 u + 3 = 0

Move

3

to the right side

2 u + 3 = 0

Subtract

3

from both sides

2 u + 3 - 3 = 0 - 3

Simplify

2 u = - 3

2 u = - 3

Divide both sides by

2

2 u = - 3

Divide both sides by

2

\frac{2 u}{2} = \frac{- 3}{2}

Simplify

u = - \frac{3}{2}

u = - \frac{3}{2}

Solve

2 u - 3 = 0 : u = \frac{3}{2}

2 u - 3 = 0

Move

3

to the right side

2 u - 3 = 0

Add

3

to both sides

2 u - 3 + 3 = 0 + 3

Simplify

2 u = 3

2 u = 3

Divide both sides by

2

2 u = 3

Divide both sides by

2

\frac{2 u}{2} = \frac{3}{2}

Simplify

u = \frac{3}{2}

u = \frac{3}{2}

The solutions to the quadratic equation are:

u = - \frac{3}{2}, u = \frac{3}{2}

Summarize in a table:

u 2 u + 3 2 u - 3 \frac{u}{( 2 u + 3 ) ( 2 u - 3 )} u < - \frac{3}{2} - - - - u = - \frac{3}{2} - 0 - Undefined - \frac{3}{2} < u < 0 - + - + u = 0 0 + - 0 0 < u < \frac{3}{2} + + - - u = \frac{3}{2} + + 0 Undefined u > \frac{3}{2} + + + +

Identify the intervals that satisfy the required condition:

> 0

- \frac{3}{2} < u < 0 or u > \frac{3}{2}

- \frac{3}{2} < u < 0 or u > \frac{3}{2}

- \frac{3}{2} < u < 0 or u > \frac{3}{2}

Substitute back

u = sin (x)

- \frac{3}{2} < sin (x) < 0 or sin (x) > \frac{3}{2}

- \frac{3}{2} < sin (x) < 0 : π + 2 πn < x < \frac{4 π}{3} + 2 πn or \frac{5 π}{3} + 2 πn < x < 2 π + 2 πn

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

a < u < b

then

a < u and u < b

- \frac{3}{2} < sin (x) and sin (x) < 0

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

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

Switch sides

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

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

Simplify

arcsin (- \frac{3}{2}) : - \frac{π}{3}

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

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{2}) = - arcsin (\frac{3}{2})

= - arcsin (\frac{3}{2})

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

arcsin (\frac{3}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{3}

= - \frac{π}{3}

Simplify

π - arcsin (- \frac{3}{2}) : \frac{4 π}{3}

π - arcsin (- \frac{3}{2})

arcsin (- \frac{3}{2}) = - \frac{π}{3}

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

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{2}) = - arcsin (\frac{3}{2})

= - arcsin (\frac{3}{2})

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

arcsin (\frac{3}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{3}

= - \frac{π}{3}

= π - (- \frac{π}{3})

Simplify

π - (- \frac{π}{3})

Apply rule

- (- a) = a

= π + \frac{π}{3}

Convert element to fraction:

π = \frac{π 3}{3}

= \frac{π 3}{3} + \frac{π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + π}{3}

Add similar elements:

3 π + π = 4 π

= \frac{4 π}{3}

= \frac{4 π}{3}

- \frac{π}{3} + 2 πn < x < \frac{4 π}{3} + 2 πn

sin (x) < 0 : - π + 2 πn < x < 2 πn

sin (x) < 0

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (0) + 2 πn < x < arcsin (0) + 2 πn

Simplify

- π - arcsin (0) : - π

- π - arcsin (0)

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= - π - 0

- π - 0 = - π

= - π

Simplify

arcsin (0) : 0

arcsin (0)

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= 0

- π + 2 πn < x < 0 + 2 πn

Simplify

- π + 2 πn < x < 2 πn

Combine the intervals

- \frac{π}{3} + 2 πn < x < \frac{4 π}{3} + 2 πn and - π + 2 πn < x < 2 πn

Merge Overlapping Intervals

π + 2 πn < x < \frac{4 π}{3} + 2 πn or \frac{5 π}{3} + 2 πn < x < 2 π + 2 πn

sin (x) > \frac{3}{2} : \frac{π}{3} + 2 πn < x < \frac{2 π}{3} + 2 πn

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

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

Simplify

arcsin (\frac{3}{2}) : \frac{π}{3}

arcsin (\frac{3}{2})

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{3}

Simplify

π - arcsin (\frac{3}{2}) : \frac{2 π}{3}

π - arcsin (\frac{3}{2})

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= π - \frac{π}{3}

Simplify

π - \frac{π}{3}

Convert element to fraction:

π = \frac{π 3}{3}

= \frac{π 3}{3} - \frac{π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π}{3}

Add similar elements:

3 π - π = 2 π

= \frac{2 π}{3}

= \frac{2 π}{3}

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

Combine the intervals

(π + 2 πn < x < \frac{4 π}{3} + 2 πn or \frac{5 π}{3} + 2 πn < x < 2 π + 2 πn) or \frac{π}{3} + 2 πn < x < \frac{2 π}{3} + 2 πn

Merge Overlapping Intervals

\frac{π}{3} + 2 πn < x < \frac{2 π}{3} + 2 πn or π + 2 πn < x < \frac{4 π}{3} + 2 πn or \frac{5 π}{3} + 2 πn < x < 2 π + 2 πn

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