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solvefor x, 3/4-sin^2(2x)>0

Solution

solve for

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

Solution

πn \leq x < \frac{π}{6} + πn or \frac{π}{3} + πn < x < \frac{2 π}{3} + πn or \frac{5 π}{6} + πn < x < π + πn

Solution steps

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

Move

\frac{3}{4}

to the right side

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

Subtract

\frac{3}{4}

from both sides

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

Simplify

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

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

Multiply both sides by

- 1

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

Multiply both sides by -1 (reverse the inequality)

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

Simplify

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

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

For

u^{n} < a

, if

n

is even then

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

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

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

a < u < b

then

a < u and u < b

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

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

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

Switch sides

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

a < u < b

then

a < u and u < b

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

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

arcsin (- \frac{3}{2}) + 2 πn < 2 x

Switch sides

2 x > arcsin (- \frac{3}{2}) + 2 πn

Simplify

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

arcsin (- \frac{3}{2}) + 2 πn

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{π}{3} + 2 πn

2 x > - \frac{π}{3} + 2 πn

Divide both sides by

2

2 x > - \frac{π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- \frac{\frac{π}{3}}{2} + \frac{2 πn}{2} : - \frac{π}{6} + πn

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= - \frac{π}{6} + πn

x > - \frac{π}{6} + πn

x > - \frac{π}{6} + πn

x > - \frac{π}{6} + πn

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

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

Simplify

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

π - arcsin (- \frac{3}{2}) + 2 πn

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}) + 2 πn

Apply rule

- (- a) = a

= π + \frac{π}{3} + 2 πn

2 x < π + \frac{π}{3} + 2 πn

Divide both sides by

2

2 x < π + \frac{π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{π}{2} + \frac{\frac{π}{3}}{2} + \frac{2 πn}{2} : \frac{π}{2} + \frac{π}{6} + πn

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{2} + \frac{π}{6} + πn

x < \frac{π}{2} + \frac{π}{6} + πn

x < \frac{π}{2} + \frac{π}{6} + πn

Simplify

\frac{π}{2} + \frac{π}{6} : \frac{2 π}{3}

\frac{π}{2} + \frac{π}{6}

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

6

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

= \frac{π 3}{6} + \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + π}{6}

Add similar elements:

3 π + π = 4 π

= \frac{4 π}{6}

Cancel the common factor:

2

= \frac{2 π}{3}

x < \frac{2 π}{3} + πn

x < \frac{2 π}{3} + πn

Combine the intervals

x > - \frac{π}{6} + πn and x < \frac{2 π}{3} + πn

Merge Overlapping Intervals

- \frac{π}{6} + πn < x < \frac{2 π}{3} + πn

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

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

a < u < b

then

a < u and u < b

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

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

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

Switch sides

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

Simplify

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

- π - arcsin (\frac{3}{2}) + 2 πn

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} + 2 πn

2 x > - π - \frac{π}{3} + 2 πn

Divide both sides by

2

2 x > - π - \frac{π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- \frac{π}{2} - \frac{\frac{π}{3}}{2} + \frac{2 πn}{2} : - \frac{π}{2} - \frac{π}{6} + πn

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= - \frac{π}{2} - \frac{π}{6} + πn

x > - \frac{π}{2} - \frac{π}{6} + πn

x > - \frac{π}{2} - \frac{π}{6} + πn

Simplify

- \frac{π}{2} - \frac{π}{6} : - \frac{2 π}{3}

- \frac{π}{2} - \frac{π}{6}

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

6

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

= - \frac{π 3}{6} - \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 - π}{6}

Add similar elements:

- 3 π - π = - 4 π

= \frac{- 4 π}{6}

Apply the fraction rule:

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

= - \frac{4 π}{6}

Cancel the common factor:

2

= - \frac{2 π}{3}

x > - \frac{2 π}{3} + πn

x > - \frac{2 π}{3} + πn

2 x < arcsin (\frac{3}{2}) + 2 πn : x < \frac{π}{6} + πn

2 x < arcsin (\frac{3}{2}) + 2 πn

Simplify

arcsin (\frac{3}{2}) + 2 πn : \frac{π}{3} + 2 πn

arcsin (\frac{3}{2}) + 2 πn

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} + 2 πn

2 x < \frac{π}{3} + 2 πn

Divide both sides by

2

2 x < \frac{π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{3}}{2} + \frac{2 πn}{2} : \frac{π}{6} + πn

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{6} + πn

x < \frac{π}{6} + πn

x < \frac{π}{6} + πn

x < \frac{π}{6} + πn

Combine the intervals

x > - \frac{2 π}{3} + πn and x < \frac{π}{6} + πn

Merge Overlapping Intervals

- \frac{2 π}{3} + πn < x < \frac{π}{6} + πn

Combine the intervals

- \frac{π}{6} + πn < x < \frac{2 π}{3} + πn and - \frac{2 π}{3} + πn < x < \frac{π}{6} + πn

Merge Overlapping Intervals

πn \leq x < \frac{π}{6} + πn or \frac{π}{3} + πn < x < \frac{2 π}{3} + πn or \frac{5 π}{6} + πn < x < π + πn

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