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Popular Trigonometry >

7.5cos(pi/6 (x+3))+10.5>13.75

Solution

7.5 cos (\frac{π}{6} (x + 3)) + 10.5 > 13.75

Solution

\frac{- 6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n < x < \frac{6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n

Interval Notation

(\frac{- 6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n, \frac{6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n)

Decimal

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

Solution steps

7.5 cos (\frac{π}{6} (x + 3)) + 10.5 > 13.75

Multiply both sides by

100

7.5 cos (\frac{π}{6} (x + 3)) + 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 cos (\frac{π}{6} (x + 3)) \cdot 100 + 10.5 \cdot 100 > 13.75 \cdot 100

Refine

750 cos (\frac{π}{6} (x + 3)) + 1050 > 1375

750 cos (\frac{π}{6} (x + 3)) + 1050 > 1375

Move

1050

to the right side

750 cos (\frac{π}{6} (x + 3)) + 1050 > 1375

Subtract

1050

from both sides

750 cos (\frac{π}{6} (x + 3)) + 1050 - 1050 > 1375 - 1050

Simplify

750 cos (\frac{π}{6} (x + 3)) > 325

750 cos (\frac{π}{6} (x + 3)) > 325

Divide both sides by

750

750 cos (\frac{π}{6} (x + 3)) > 325

Divide both sides by

750

\frac{750 cos ( \frac{π}{6} ( x + 3 ) )}{750} > \frac{325}{750}

Simplify

cos (\frac{π}{6} (x + 3)) > \frac{13}{30}

cos (\frac{π}{6} (x + 3)) > \frac{13}{30}

For

cos (x) > a

, if

- 1 \leq a < 1

then

- arccos (a) + 2 πn < x < arccos (a) + 2 πn

- arccos (\frac{13}{30}) + 2 πn < \frac{π}{6} (x + 3) < arccos (\frac{13}{30}) + 2 πn

a < u < b

then

a < u and u < b

- arccos (\frac{13}{30}) + 2 πn < \frac{π}{6} (x + 3) and \frac{π}{6} (x + 3) < arccos (\frac{13}{30}) + 2 πn

- arccos (\frac{13}{30}) + 2 πn < \frac{π}{6} (x + 3) : x > \frac{- 6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n

- arccos (\frac{13}{30}) + 2 πn < \frac{π}{6} (x + 3)

Switch sides

\frac{π}{6} (x + 3) > - arccos (\frac{13}{30}) + 2 πn

Multiply both sides by

6

\frac{π}{6} (x + 3) > - arccos (\frac{13}{30}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} (x + 3) > - 6 arccos (\frac{13}{30}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} (x + 3) > - 6 arccos (\frac{13}{30}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} (x + 3) : π (x + 3)

6 \cdot \frac{π}{6} (x + 3)

Multiply fractions:

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

= \frac{6 π}{6} (x + 3)

Cancel the common factor:

6

= (x + 3) π

Simplify

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

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

Multiply the numbers:

6 \cdot 2 = 12

= - 6 arccos (\frac{13}{30}) + 12 πn

π (x + 3) > - 6 arccos (\frac{13}{30}) + 12 πn

π (x + 3) > - 6 arccos (\frac{13}{30}) + 12 πn

π (x + 3) > - 6 arccos (\frac{13}{30}) + 12 πn

Divide both sides by

π

π (x + 3) > - 6 arccos (\frac{13}{30}) + 12 πn

Divide both sides by

π

\frac{π ( x + 3 )}{π} > - \frac{6 arccos ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Simplify

x + 3 > - \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n

x + 3 > - \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n

Move

3

to the right side

x + 3 > - \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n

Subtract

3

from both sides

x + 3 - 3 > - \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n - 3

Simplify

x > - \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n - 3

x > - \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n - 3

Simplify

- \frac{6 arccos ( \frac{13}{30} )}{π} - 3 : \frac{- 6 arccos ( \frac{13}{30} ) - 3 π}{π}

- \frac{6 arccos ( \frac{13}{30} )}{π} - 3

Convert element to fraction:

3 = \frac{3 π}{π}

= - \frac{6 arccos ( \frac{13}{30} )}{π} - \frac{3 π}{π}

Since the denominators are equal, combine the fractions:

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

= \frac{- 6 arccos ( \frac{13}{30} ) - 3 π}{π}

x > \frac{- 6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n

\frac{π}{6} (x + 3) < arccos (\frac{13}{30}) + 2 πn : x < \frac{6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n

\frac{π}{6} (x + 3) < arccos (\frac{13}{30}) + 2 πn

Multiply both sides by

6

\frac{π}{6} (x + 3) < arccos (\frac{13}{30}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} (x + 3) < 6 arccos (\frac{13}{30}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} (x + 3) < 6 arccos (\frac{13}{30}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} (x + 3) : π (x + 3)

6 \cdot \frac{π}{6} (x + 3)

Multiply fractions:

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

= \frac{6 π}{6} (x + 3)

Cancel the common factor:

6

= (x + 3) π

Simplify

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

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

Multiply the numbers:

6 \cdot 2 = 12

= 6 arccos (\frac{13}{30}) + 12 πn

π (x + 3) < 6 arccos (\frac{13}{30}) + 12 πn

π (x + 3) < 6 arccos (\frac{13}{30}) + 12 πn

π (x + 3) < 6 arccos (\frac{13}{30}) + 12 πn

Divide both sides by

π

π (x + 3) < 6 arccos (\frac{13}{30}) + 12 πn

Divide both sides by

π

\frac{π ( x + 3 )}{π} < \frac{6 arccos ( \frac{13}{30} )}{π} + \frac{12 πn}{π}

Simplify

x + 3 < \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n

x + 3 < \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n

Move

3

to the right side

x + 3 < \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n

Subtract

3

from both sides

x + 3 - 3 < \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n - 3

Simplify

x < \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n - 3

x < \frac{6 arccos ( \frac{13}{30} )}{π} + 12 n - 3

Simplify

\frac{6 arccos ( \frac{13}{30} )}{π} - 3 : \frac{6 arccos ( \frac{13}{30} ) - 3 π}{π}

\frac{6 arccos ( \frac{13}{30} )}{π} - 3

Convert element to fraction:

3 = \frac{3 π}{π}

= \frac{6 arccos ( \frac{13}{30} )}{π} - \frac{3 π}{π}

Since the denominators are equal, combine the fractions:

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

= \frac{6 arccos ( \frac{13}{30} ) - 3 π}{π}

x < \frac{6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n

Combine the intervals

x > \frac{- 6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n and x < \frac{6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n

Merge Overlapping Intervals

\frac{- 6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n < x < \frac{6 arccos ( \frac{13}{30} ) - 3 π}{π} + 12 n

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