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-7.5sin(pi/6 x)+10.5>13.75

Solution

- 7.5 sin (\frac{π}{6} x) + 10.5 > 13.75

Solution

\frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n < x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

Interval Notation

(\frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n, - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n)

Decimal Notation

- 5.14402 \dots + 12 n < x < - 0.85597 \dots + 12 n

Solution steps

- 7.5 sin (\frac{π}{6} x) + 10.5 > 13.75

Multiply both sides by

100

- 7.5 sin (\frac{π}{6} x) + 10.5 > 13.75

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

- 7.5 sin (\frac{π}{6} x) \cdot 100 + 10.5 \cdot 100 > 13.75 \cdot 100

Refine

- 750 sin (\frac{π}{6} x) + 1050 > 1375

- 750 sin (\frac{π}{6} x) + 1050 > 1375

Move

1050

to the right side

- 750 sin (\frac{π}{6} x) + 1050 > 1375

Subtract

1050

from both sides

- 750 sin (\frac{π}{6} x) + 1050 - 1050 > 1375 - 1050

Simplify

- 750 sin (\frac{π}{6} x) > 325

- 750 sin (\frac{π}{6} x) > 325

Multiply both sides by

- 1

- 750 sin (\frac{π}{6} x) > 325

Multiply both sides by -1 (reverse the inequality)

(- 750 sin (\frac{π}{6} x)) (- 1) < 325 (- 1)

Simplify

750 sin (\frac{π}{6} x) < - 325

750 sin (\frac{π}{6} x) < - 325

Divide both sides by

750

750 sin (\frac{π}{6} x) < - 325

Divide both sides by

750

\frac{750 sin ( \frac{π}{6} x )}{750} < \frac{- 325}{750}

Simplify

\frac{750 sin ( \frac{π}{6} x )}{750} < \frac{- 325}{750}

Simplify

\frac{750 sin ( \frac{π}{6} x )}{750} : sin (\frac{π}{6} x)

\frac{750 sin ( \frac{π}{6} x )}{750}

Divide the numbers:

\frac{750}{750} = 1

= sin (\frac{π}{6} x)

Simplify

\frac{- 325}{750} : - \frac{13}{30}

\frac{- 325}{750}

Apply the fraction rule:

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

= - \frac{325}{750}

Cancel the common factor:

25

= - \frac{13}{30}

sin (\frac{π}{6} x) < - \frac{13}{30}

sin (\frac{π}{6} x) < - \frac{13}{30}

sin (\frac{π}{6} x) < - \frac{13}{30}

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (- \frac{13}{30}) + 2 πn < \frac{π}{6} x < arcsin (- \frac{13}{30}) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (- \frac{13}{30}) + 2 πn < \frac{π}{6} x and \frac{π}{6} x < arcsin (- \frac{13}{30}) + 2 πn

- π - arcsin (- \frac{13}{30}) + 2 πn < \frac{π}{6} x : x > \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n

- π - arcsin (- \frac{13}{30}) + 2 πn < \frac{π}{6} x

Switch sides

\frac{π}{6} x > - π - arcsin (- \frac{13}{30}) + 2 πn

Simplify

- π - arcsin (- \frac{13}{30}) + 2 πn : - π + arcsin (\frac{13}{30}) + 2 πn

- π - arcsin (- \frac{13}{30}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{13}{30}) = - arcsin (\frac{13}{30})

= - π - (- arcsin (\frac{13}{30})) + 2 πn

Apply rule

- (- a) = a

= - π + arcsin (\frac{13}{30}) + 2 πn

\frac{π}{6} x > - π + arcsin (\frac{13}{30}) + 2 πn

Multiply both sides by

6

\frac{π}{6} x > - π + arcsin (\frac{13}{30}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x > - 6 π + 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x > - 6 π + 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiply fractions:

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

= \frac{6 π}{6} x

Cancel the common factor:

6

= x π

Simplify

- 6 π + 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn : - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

- 6 π + 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

π x > - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

π x > - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

π x > - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

Divide both sides by

π

π x > - 6 π + 6 arcsin (\frac{13}{30}) + 12 πn

Divide both sides by

π

\frac{π x}{π} > - \frac{6 π}{π} + \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} > - \frac{6 π}{π} + \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

- \frac{6 π}{π} + \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π} : - 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

- \frac{6 π}{π} + \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Cancel

\frac{6 π}{π} : 6

\frac{6 π}{π}

Cancel the common factor:

π

= 6

= - 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= - 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

x > - 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

x > - 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

Simplify

- 6 + \frac{6 arcsin ( \frac{13}{30} )}{π} : \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π}

- 6 + \frac{6 arcsin ( \frac{13}{30} )}{π}

Convert element to fraction:

6 = \frac{6 π}{π}

= - \frac{6 π}{π} + \frac{6 arcsin ( \frac{13}{30} )}{π}

Since the denominators are equal, combine the fractions:

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

= \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π}

x > \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n

x > \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n

\frac{π}{6} x < arcsin (- \frac{13}{30}) + 2 πn : x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

\frac{π}{6} x < arcsin (- \frac{13}{30}) + 2 πn

Simplify

arcsin (- \frac{13}{30}) + 2 πn : - arcsin (\frac{13}{30}) + 2 πn

arcsin (- \frac{13}{30}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{13}{30}) = - arcsin (\frac{13}{30})

= - arcsin (\frac{13}{30}) + 2 πn

\frac{π}{6} x < - arcsin (\frac{13}{30}) + 2 πn

Multiply both sides by

6

\frac{π}{6} x < - arcsin (\frac{13}{30}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x < - 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x < - 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiply fractions:

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

= \frac{6 π}{6} x

Cancel the common factor:

6

= x π

Simplify

- 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn : - 6 arcsin (\frac{13}{30}) + 12 πn

- 6 arcsin (\frac{13}{30}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= - 6 arcsin (\frac{13}{30}) + 12 πn

π x < - 6 arcsin (\frac{13}{30}) + 12 πn

π x < - 6 arcsin (\frac{13}{30}) + 12 πn

π x < - 6 arcsin (\frac{13}{30}) + 12 πn

Divide both sides by

π

π x < - 6 arcsin (\frac{13}{30}) + 12 πn

Divide both sides by

π

\frac{π x}{π} < - \frac{6 arcsin ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Simplify

x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

Combine the intervals

x > \frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n and x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

Merge Overlapping Intervals

\frac{- 6 π + 6 arcsin ( \frac{13}{30} )}{π} + 12 n < x < - \frac{6 arcsin ( \frac{13}{30} )}{π} + 12 n

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