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Popular Trigonometry >

1/((sin(x))^2)< 4/3 ,0<x<12

Solution

\frac{1}{( sin ( x ) ) ^{2}} < \frac{4}{3}, 0^{\circ} < x < 1 2^{\circ}

Solution

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

Interval Notation

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

Decimal

1.04719 \dots + 2 πn < x < 2.09439 \dots + 2 πn or - 2.09439 \dots + 2 πn < x < - 1.04719 \dots + 2 πn

Solution steps

\frac{1}{( sin ( x ) ) ^{2}} < \frac{4}{3}, 0^{\circ} < x < 1 2^{\circ}

Rewrite in standard form

\frac{1}{sin ^{2} ( x )} < \frac{4}{3}

Subtract

\frac{4}{3}

from both sides

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

Simplify

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

Simplify

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

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

Least Common Multiplier of

sin^{2} (x), 3 : 3 sin^{2} (x)

sin^{2} (x), 3

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin^{2} (x)

3

= 3 sin^{2} (x)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

3 sin^{2} (x)

For

\frac{1}{sin ^{2} ( x )} :

multiply the denominator and numerator by

3

\frac{1}{sin ^{2} ( x )} = \frac{1 \cdot 3}{sin ^{2} ( x ) \cdot 3} = \frac{3}{3 sin ^{2} ( x )}

For

\frac{4}{3} :

multiply the denominator and numerator by

sin^{2} (x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

Multiply both sides by

3

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

Simplify

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

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

Factor

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

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

Factor

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

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

Factor out common term

- 1

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

Factor

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

4 sin^{2} (x) - 3

Rewrite

4 sin^{2} (x) - 3

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

4 sin^{2} (x) - 3

Rewrite

4

2^{2}

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

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

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

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

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( 2 sin ( x ) + 3 ) ( 2 sin ( x ) - 3 ) ) ( - 1 )}{sin ^{2} ( x )} > 0 \cdot (- 1)

Simplify

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

Identify the intervals

Find the signs of the factors of

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

Find the signs of

2 sin (x) + 3

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

2 sin (x) + 3 = 0

Move

3

to the right side

2 sin (x) + 3 = 0

Subtract

3

from both sides

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

Simplify

2 sin (x) = - 3

2 sin (x) = - 3

Divide both sides by

2

2 sin (x) = - 3

Divide both sides by

2

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

Simplify

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

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

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

2 sin (x) + 3 < 0

Move

3

to the right side

2 sin (x) + 3 < 0

Subtract

3

from both sides

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

Simplify

2 sin (x) < - 3

2 sin (x) < - 3

Divide both sides by

2

2 sin (x) < - 3

Divide both sides by

2

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

Simplify

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

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

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

2 sin (x) + 3 > 0

Move

3

to the right side

2 sin (x) + 3 > 0

Subtract

3

from both sides

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

Simplify

2 sin (x) > - 3

2 sin (x) > - 3

Divide both sides by

2

2 sin (x) > - 3

Divide both sides by

2

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

Simplify

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

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

Find the signs of

2 sin (x) - 3

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

2 sin (x) - 3 = 0

Move

3

to the right side

2 sin (x) - 3 = 0

Add

3

to both sides

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

Simplify

2 sin (x) = 3

2 sin (x) = 3

Divide both sides by

2

2 sin (x) = 3

Divide both sides by

2

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

Simplify

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

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

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

2 sin (x) - 3 < 0

Move

3

to the right side

2 sin (x) - 3 < 0

Add

3

to both sides

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

Simplify

2 sin (x) < 3

2 sin (x) < 3

Divide both sides by

2

2 sin (x) < 3

Divide both sides by

2

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

Simplify

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

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

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

2 sin (x) - 3 > 0

Move

3

to the right side

2 sin (x) - 3 > 0

Add

3

to both sides

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

Simplify

2 sin (x) > 3

2 sin (x) > 3

Divide both sides by

2

2 sin (x) > 3

Divide both sides by

2

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

Simplify

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

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

Find the signs of

sin^{2} (x)

sin^{2} (x) = 0 : sin (x) = 0

sin^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

sin (x) = 0

sin^{2} (x) > 0 : sin (x) < 0 or sin (x) > 0

sin^{2} (x) > 0

For

u^{n} > 0

, if

n

is even then

u < 0

u > 0

sin (x) < 0 or sin (x) > 0

Find singularity points

Find the zeros of the denominator

sin^{2} (x) :

No Solution

sin^{2} (x) = 0

The sides are not equal

No Solution

Summarize in a table:

2 sin (x) + 3 2 sin (x) - 3 sin^{2} (x) \frac{( 2 s i n ( x ) + 3 ) ( 2 s i n ( x ) - 3 )}{s i n ^{2} ( x )} sin (x) < - \frac{3}{2} - - + + sin (x) = - \frac{3}{2} 0 - + 0 - \frac{3}{2} < sin (x) < 0 + - + - sin (x) = 0 + - 0 Undefined 0 < sin (x) < \frac{3}{2} + - + - sin (x) = \frac{3}{2} + 0 + 0 sin (x) > \frac{3}{2} + + + +

Identify the intervals that satisfy the required condition:

> 0

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

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

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

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

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

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

Simplify

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

= \frac{- 2 π}{3}

Apply the fraction rule:

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

= - \frac{2 π}{3}

= - \frac{2 π}{3}

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}

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

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

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