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(0.75sin((2pix)/3))+1.25<1.8

Solution

(0.75 \cdot sin (\frac{2 π \cdot x}{3})) + 1.25 < 1.8

Solution

\frac{- 3 π - 3 arcsin ( \frac{11}{15} )}{2 π} + 3 n < x < \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

Interval Notation

(\frac{- 3 π - 3 arcsin ( \frac{11}{15} )}{2 π} + 3 n, \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n)

Decimal Notation

- 1.89305 \dots + 3 n < x < 0.39305 \dots + 3 n

Solution steps

0.75 sin (\frac{2 π x}{3}) + 1.25 < 1.8

Multiply both sides by

100

0.75 sin (\frac{2 π x}{3}) + 1.25 < 1.8

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.75 sin (\frac{2 π x}{3}) \cdot 100 + 1.25 \cdot 100 < 1.8 \cdot 100

Refine

75 sin (\frac{2 π x}{3}) + 125 < 180

75 sin (\frac{2 π x}{3}) + 125 < 180

Move

125

to the right side

75 sin (\frac{2 π x}{3}) + 125 < 180

Subtract

125

from both sides

75 sin (\frac{2 π x}{3}) + 125 - 125 < 180 - 125

Simplify

75 sin (\frac{2 π x}{3}) < 55

75 sin (\frac{2 π x}{3}) < 55

Divide both sides by

75

75 sin (\frac{2 π x}{3}) < 55

Divide both sides by

75

\frac{75 sin ( \frac{2 π x}{3} )}{75} < \frac{55}{75}

Simplify

sin (\frac{2 π x}{3}) < \frac{11}{15}

sin (\frac{2 π x}{3}) < \frac{11}{15}

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (\frac{11}{15}) + 2 πn < \frac{2 π x}{3} < arcsin (\frac{11}{15}) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (\frac{11}{15}) + 2 πn < \frac{2 π x}{3} and \frac{2 π x}{3} < arcsin (\frac{11}{15}) + 2 πn

- π - arcsin (\frac{11}{15}) + 2 πn < \frac{2 π x}{3} : x > \frac{- 3 π - 3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

- π - arcsin (\frac{11}{15}) + 2 πn < \frac{2 π x}{3}

Switch sides

\frac{2 π x}{3} > - π - arcsin (\frac{11}{15}) + 2 πn

Multiply both sides by

3

\frac{2 π x}{3} > - π - arcsin (\frac{11}{15}) + 2 πn

Multiply both sides by

3

\frac{3 \cdot 2 π x}{3} > - 3 π - 3 arcsin (\frac{11}{15}) + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 π x}{3} > - 3 π - 3 arcsin (\frac{11}{15}) + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 π x}{3} : 2 π x

\frac{3 \cdot 2 π x}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π x}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 π x

Simplify

- 3 π - 3 arcsin (\frac{11}{15}) + 3 \cdot 2 πn : - 3 π - 3 arcsin (\frac{11}{15}) + 6 πn

- 3 π - 3 arcsin (\frac{11}{15}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= - 3 π - 3 arcsin (\frac{11}{15}) + 6 πn

2 π x > - 3 π - 3 arcsin (\frac{11}{15}) + 6 πn

2 π x > - 3 π - 3 arcsin (\frac{11}{15}) + 6 πn

2 π x > - 3 π - 3 arcsin (\frac{11}{15}) + 6 πn

Divide both sides by

2 π

2 π x > - 3 π - 3 arcsin (\frac{11}{15}) + 6 πn

Divide both sides by

2 π

\frac{2 π x}{2 π} > - \frac{3 π}{2 π} - \frac{3 arcsin ( \frac{11}{15} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π x}{2 π} > - \frac{3 π}{2 π} - \frac{3 arcsin ( \frac{11}{15} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π x}{2 π} : x

\frac{2 π x}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

- \frac{3 π}{2 π} - \frac{3 arcsin ( \frac{11}{15} )}{2 π} + \frac{6 πn}{2 π} : - \frac{3}{2} - \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

- \frac{3 π}{2 π} - \frac{3 arcsin ( \frac{11}{15} )}{2 π} + \frac{6 πn}{2 π}

Cancel

\frac{3 π}{2 π} : \frac{3}{2}

\frac{3 π}{2 π}

Cancel the common factor:

π

= \frac{3}{2}

= - \frac{3}{2} - \frac{3 arcsin ( \frac{11}{15} )}{2 π} + \frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Divide the numbers:

\frac{6}{2} = 3

= \frac{3 πn}{π}

Cancel the common factor:

π

= 3 n

= 3 n

= - \frac{3}{2} - \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

x > - \frac{3}{2} - \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

x > - \frac{3}{2} - \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

Simplify

- \frac{3}{2} - \frac{3 arcsin ( \frac{11}{15} )}{2 π} : \frac{- 3 π - 3 arcsin ( \frac{11}{15} )}{2 π}

- \frac{3}{2} - \frac{3 arcsin ( \frac{11}{15} )}{2 π}

Least Common Multiplier of

2, 2 π : 2 π

2, 2 π

Lowest Common Multiplier (LCM)

Least Common Multiplier of

2, 2 : 2

2, 2

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

2

2

= 2

Multiply the numbers:

2 = 2

= 2

Compute an expression comprised of factors that appear either in

2

2 π

= 2 π

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

2 π

For

\frac{3}{2} :

multiply the denominator and numerator by

π

\frac{3}{2} = \frac{3 π}{2 π}

= - \frac{3 π}{2 π} - \frac{3 arcsin ( \frac{11}{15} )}{2 π}

Since the denominators are equal, combine the fractions:

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

= \frac{- 3 π - 3 arcsin ( \frac{11}{15} )}{2 π}

x > \frac{- 3 π - 3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

x > \frac{- 3 π - 3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

\frac{2 π x}{3} < arcsin (\frac{11}{15}) + 2 πn : x < \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

\frac{2 π x}{3} < arcsin (\frac{11}{15}) + 2 πn

Multiply both sides by

3

\frac{2 π x}{3} < arcsin (\frac{11}{15}) + 2 πn

Multiply both sides by

3

\frac{3 \cdot 2 π x}{3} < 3 arcsin (\frac{11}{15}) + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 π x}{3} < 3 arcsin (\frac{11}{15}) + 3 \cdot 2 πn

Simplify

\frac{3 \cdot 2 π x}{3} : 2 π x

\frac{3 \cdot 2 π x}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π x}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 π x

Simplify

3 arcsin (\frac{11}{15}) + 3 \cdot 2 πn : 3 arcsin (\frac{11}{15}) + 6 πn

3 arcsin (\frac{11}{15}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 3 arcsin (\frac{11}{15}) + 6 πn

2 π x < 3 arcsin (\frac{11}{15}) + 6 πn

2 π x < 3 arcsin (\frac{11}{15}) + 6 πn

2 π x < 3 arcsin (\frac{11}{15}) + 6 πn

Divide both sides by

2 π

2 π x < 3 arcsin (\frac{11}{15}) + 6 πn

Divide both sides by

2 π

\frac{2 π x}{2 π} < \frac{3 arcsin ( \frac{11}{15} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π x}{2 π} < \frac{3 arcsin ( \frac{11}{15} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π x}{2 π} : x

\frac{2 π x}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{3 arcsin ( \frac{11}{15} )}{2 π} + \frac{6 πn}{2 π} : \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

\frac{3 arcsin ( \frac{11}{15} )}{2 π} + \frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Cancel

\frac{6 πn}{2 π} : 3 n

\frac{6 πn}{2 π}

Divide the numbers:

\frac{6}{2} = 3

= \frac{3 πn}{π}

Cancel the common factor:

π

= 3 n

= 3 n

= \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

x < \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

x < \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

x < \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

Combine the intervals

x > \frac{- 3 π - 3 arcsin ( \frac{11}{15} )}{2 π} + 3 n and x < \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

Merge Overlapping Intervals

\frac{- 3 π - 3 arcsin ( \frac{11}{15} )}{2 π} + 3 n < x < \frac{3 arcsin ( \frac{11}{15} )}{2 π} + 3 n

Popular Examples

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(0.75*sin((2pi*x)/3))+1.25<1.8

Solution

Solution

Popular Examples

(0.75sin((2pix)/3))+1.25<1.8