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Popular Trigonometry >

10<5sin(pi/6 (x-10))+6

Solution

10 < 5 sin (\frac{π}{6} (x - 10)) + 6

Solution

\frac{6 arcsin ( \frac{4}{5} ) + 10 π}{π} + 12 n < x < \frac{16 π - 6 arcsin ( \frac{4}{5} )}{π} + 12 n

Interval Notation

(\frac{6 arcsin ( \frac{4}{5} ) + 10 π}{π} + 12 n, \frac{16 π - 6 arcsin ( \frac{4}{5} )}{π} + 12 n)

Decimal

11.77100 \dots + 12 n < x < 14.22899 \dots + 12 n

Solution steps

10 < 5 sin (\frac{π}{6} (x - 10)) + 6

Switch sides

5 sin (\frac{π}{6} (x - 10)) + 6 > 10

Move

6

to the right side

5 sin (\frac{π}{6} (x - 10)) + 6 > 10

Subtract

6

from both sides

5 sin (\frac{π}{6} (x - 10)) + 6 - 6 > 10 - 6

Simplify

5 sin (\frac{π}{6} (x - 10)) > 4

5 sin (\frac{π}{6} (x - 10)) > 4

Divide both sides by

5

5 sin (\frac{π}{6} (x - 10)) > 4

Divide both sides by

5

\frac{5 sin ( \frac{π}{6} ( x - 10 ) )}{5} > \frac{4}{5}

Simplify

sin (\frac{π}{6} (x - 10)) > \frac{4}{5}

sin (\frac{π}{6} (x - 10)) > \frac{4}{5}

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (\frac{4}{5}) + 2 πn < \frac{π}{6} (x - 10) < π - arcsin (\frac{4}{5}) + 2 πn

a < u < b

then

a < u and u < b

arcsin (\frac{4}{5}) + 2 πn < \frac{π}{6} (x - 10) and \frac{π}{6} (x - 10) < π - arcsin (\frac{4}{5}) + 2 πn

arcsin (\frac{4}{5}) + 2 πn < \frac{π}{6} (x - 10) : x > \frac{6 arcsin ( \frac{4}{5} ) + 10 π}{π} + 12 n

arcsin (\frac{4}{5}) + 2 πn < \frac{π}{6} (x - 10)

Switch sides

\frac{π}{6} (x - 10) > arcsin (\frac{4}{5}) + 2 πn

Multiply both sides by

6

\frac{π}{6} (x - 10) > arcsin (\frac{4}{5}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} (x - 10) > 6 arcsin (\frac{4}{5}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} (x - 10) > 6 arcsin (\frac{4}{5}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} (x - 10) : π (x - 10)

6 \cdot \frac{π}{6} (x - 10)

Multiply fractions:

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

= \frac{6 π}{6} (x - 10)

Cancel the common factor:

6

= (x - 10) π

Simplify

6 arcsin (\frac{4}{5}) + 6 \cdot 2 πn : 6 arcsin (\frac{4}{5}) + 12 πn

6 arcsin (\frac{4}{5}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 6 arcsin (\frac{4}{5}) + 12 πn

π (x - 10) > 6 arcsin (\frac{4}{5}) + 12 πn

π (x - 10) > 6 arcsin (\frac{4}{5}) + 12 πn

π (x - 10) > 6 arcsin (\frac{4}{5}) + 12 πn

Divide both sides by

π

π (x - 10) > 6 arcsin (\frac{4}{5}) + 12 πn

Divide both sides by

π

\frac{π ( x - 10 )}{π} > \frac{6 arcsin ( \frac{4}{5} )}{π} + \frac{12 πn}{π}

Simplify

x - 10 > \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n

x - 10 > \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n

Move

10

to the right side

x - 10 > \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n

Add

10

to both sides

x - 10 + 10 > \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n + 10

Simplify

x > \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n + 10

x > \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n + 10

Simplify

\frac{6 arcsin ( \frac{4}{5} )}{π} + 10 : \frac{6 arcsin ( \frac{4}{5} ) + 10 π}{π}

\frac{6 arcsin ( \frac{4}{5} )}{π} + 10

Convert element to fraction:

10 = \frac{10 π}{π}

= \frac{6 arcsin ( \frac{4}{5} )}{π} + \frac{10 π}{π}

Since the denominators are equal, combine the fractions:

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

= \frac{6 arcsin ( \frac{4}{5} ) + 10 π}{π}

x > \frac{6 arcsin ( \frac{4}{5} ) + 10 π}{π} + 12 n

\frac{π}{6} (x - 10) < π - arcsin (\frac{4}{5}) + 2 πn : x < \frac{16 π - 6 arcsin ( \frac{4}{5} )}{π} + 12 n

\frac{π}{6} (x - 10) < π - arcsin (\frac{4}{5}) + 2 πn

Multiply both sides by

6

\frac{π}{6} (x - 10) < π - arcsin (\frac{4}{5}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} (x - 10) < 6 π - 6 arcsin (\frac{4}{5}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} (x - 10) < 6 π - 6 arcsin (\frac{4}{5}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} (x - 10) : π (x - 10)

6 \cdot \frac{π}{6} (x - 10)

Multiply fractions:

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

= \frac{6 π}{6} (x - 10)

Cancel the common factor:

6

= (x - 10) π

Simplify

6 π - 6 arcsin (\frac{4}{5}) + 6 \cdot 2 πn : 6 π - 6 arcsin (\frac{4}{5}) + 12 πn

6 π - 6 arcsin (\frac{4}{5}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 6 π - 6 arcsin (\frac{4}{5}) + 12 πn

π (x - 10) < 6 π - 6 arcsin (\frac{4}{5}) + 12 πn

π (x - 10) < 6 π - 6 arcsin (\frac{4}{5}) + 12 πn

π (x - 10) < 6 π - 6 arcsin (\frac{4}{5}) + 12 πn

Divide both sides by

π

π (x - 10) < 6 π - 6 arcsin (\frac{4}{5}) + 12 πn

Divide both sides by

π

\frac{π ( x - 10 )}{π} < \frac{6 π}{π} - \frac{6 arcsin ( \frac{4}{5} )}{π} + \frac{12 πn}{π}

Simplify

\frac{π ( x - 10 )}{π} < \frac{6 π}{π} - \frac{6 arcsin ( \frac{4}{5} )}{π} + \frac{12 πn}{π}

Simplify

\frac{π ( x - 10 )}{π} : x - 10

\frac{π ( x - 10 )}{π}

Cancel the common factor:

π

= x - 10

Simplify

\frac{6 π}{π} - \frac{6 arcsin ( \frac{4}{5} )}{π} + \frac{12 πn}{π} : 6 - \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n

\frac{6 π}{π} - \frac{6 arcsin ( \frac{4}{5} )}{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{4}{5} )}{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{4}{5} )}{π} + 12 n

x - 10 < 6 - \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n

x - 10 < 6 - \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n

x - 10 < 6 - \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n

Move

10

to the right side

x - 10 < 6 - \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n

Add

10

to both sides

x - 10 + 10 < 6 - \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n + 10

Simplify

x - 10 + 10 < 6 - \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n + 10

Simplify

x - 10 + 10 : x

x - 10 + 10

Add similar elements:

- 10 + 10 < 0

= x

Simplify

6 - \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n + 10 : 12 n + 16 - \frac{6 arcsin ( \frac{4}{5} )}{π}

6 - \frac{6 arcsin ( \frac{4}{5} )}{π} + 12 n + 10

Add the numbers:

6 + 10 = 16

= 12 n + 16 - \frac{6 arcsin ( \frac{4}{5} )}{π}

x < 12 n + 16 - \frac{6 arcsin ( \frac{4}{5} )}{π}

x < 12 n + 16 - \frac{6 arcsin ( \frac{4}{5} )}{π}

x < 12 n + 16 - \frac{6 arcsin ( \frac{4}{5} )}{π}

Simplify

16 - \frac{6 arcsin ( \frac{4}{5} )}{π} : \frac{16 π - 6 arcsin ( \frac{4}{5} )}{π}

16 - \frac{6 arcsin ( \frac{4}{5} )}{π}

Convert element to fraction:

16 = \frac{16 π}{π}

= \frac{16 π}{π} - \frac{6 arcsin ( \frac{4}{5} )}{π}

Since the denominators are equal, combine the fractions:

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

= \frac{16 π - 6 arcsin ( \frac{4}{5} )}{π}

x < \frac{16 π - 6 arcsin ( \frac{4}{5} )}{π} + 12 n

Combine the intervals

x > \frac{6 arcsin ( \frac{4}{5} ) + 10 π}{π} + 12 n and x < \frac{16 π - 6 arcsin ( \frac{4}{5} )}{π} + 12 n

Merge Overlapping Intervals

\frac{6 arcsin ( \frac{4}{5} ) + 10 π}{π} + 12 n < x < \frac{16 π - 6 arcsin ( \frac{4}{5} )}{π} + 12 n

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