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Popular Trigonometry >

1250>600cos((2pi)/3 (t-1))+1000

Solution

1250 > 600 cos (\frac{2 π}{3} (t - 1)) + 1000

Solution

\frac{3 arccos ( \frac{5}{12} ) + 2 π}{2 π} + 3 n < t < \frac{8 π - 3 arccos ( \frac{5}{12} )}{2 π} + 3 n

Interval Notation

(\frac{3 arccos ( \frac{5}{12} ) + 2 π}{2 π} + 3 n, \frac{8 π - 3 arccos ( \frac{5}{12} )}{2 π} + 3 n)

Decimal

1.54479 \dots + 3 n < t < 3.45520 \dots + 3 n

Solution steps

1250 > 600 cos (\frac{2 π}{3} (t - 1)) + 1000

Switch sides

600 cos (\frac{2 π}{3} (t - 1)) + 1000 < 1250

Move

1000

to the right side

600 cos (\frac{2 π}{3} (t - 1)) + 1000 < 1250

Subtract

1000

from both sides

600 cos (\frac{2 π}{3} (t - 1)) + 1000 - 1000 < 1250 - 1000

Simplify

600 cos (\frac{2 π}{3} (t - 1)) < 250

600 cos (\frac{2 π}{3} (t - 1)) < 250

Divide both sides by

600

600 cos (\frac{2 π}{3} (t - 1)) < 250

Divide both sides by

600

\frac{600 cos ( \frac{2 π}{3} ( t - 1 ) )}{600} < \frac{250}{600}

Simplify

cos (\frac{2 π}{3} (t - 1)) < \frac{5}{12}

cos (\frac{2 π}{3} (t - 1)) < \frac{5}{12}

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

arccos (\frac{5}{12}) + 2 πn < \frac{2 π}{3} (t - 1) < 2 π - arccos (\frac{5}{12}) + 2 πn

a < u < b

then

a < u and u < b

arccos (\frac{5}{12}) + 2 πn < \frac{2 π}{3} (t - 1) and \frac{2 π}{3} (t - 1) < 2 π - arccos (\frac{5}{12}) + 2 πn

arccos (\frac{5}{12}) + 2 πn < \frac{2 π}{3} (t - 1) : t > \frac{3 arccos ( \frac{5}{12} ) + 2 π}{2 π} + 3 n

arccos (\frac{5}{12}) + 2 πn < \frac{2 π}{3} (t - 1)

Switch sides

\frac{2 π}{3} (t - 1) > arccos (\frac{5}{12}) + 2 πn

Multiply both sides by

3

\frac{2 π}{3} (t - 1) > arccos (\frac{5}{12}) + 2 πn

Multiply both sides by

3

3 \cdot \frac{2 π}{3} (t - 1) > 3 arccos (\frac{5}{12}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{2 π}{3} (t - 1) > 3 arccos (\frac{5}{12}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{2 π}{3} (t - 1) : 2 π (t - 1)

3 \cdot \frac{2 π}{3} (t - 1)

Multiply fractions:

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

= \frac{2 \cdot 3 π}{3} (t - 1)

Cancel the common factor:

3

= (t - 1) \cdot 2 π

Simplify

3 arccos (\frac{5}{12}) + 3 \cdot 2 πn : 3 arccos (\frac{5}{12}) + 6 πn

3 arccos (\frac{5}{12}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 3 arccos (\frac{5}{12}) + 6 πn

2 π (t - 1) > 3 arccos (\frac{5}{12}) + 6 πn

2 π (t - 1) > 3 arccos (\frac{5}{12}) + 6 πn

2 π (t - 1) > 3 arccos (\frac{5}{12}) + 6 πn

Divide both sides by

2 π

2 π (t - 1) > 3 arccos (\frac{5}{12}) + 6 πn

Divide both sides by

2 π

\frac{2 π ( t - 1 )}{2 π} > \frac{3 arccos ( \frac{5}{12} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π ( t - 1 )}{2 π} > \frac{3 arccos ( \frac{5}{12} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π ( t - 1 )}{2 π} : t - 1

\frac{2 π ( t - 1 )}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π ( t - 1 )}{π}

Cancel the common factor:

π

= t - 1

Simplify

\frac{3 arccos ( \frac{5}{12} )}{2 π} + \frac{6 πn}{2 π} : \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n

\frac{3 arccos ( \frac{5}{12} )}{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 arccos ( \frac{5}{12} )}{2 π} + 3 n

t - 1 > \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n

t - 1 > \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n

t - 1 > \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n

Move

1

to the right side

t - 1 > \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n

Add

1

to both sides

t - 1 + 1 > \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n + 1

Simplify

t > \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n + 1

t > \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n + 1

Simplify

\frac{3 arccos ( \frac{5}{12} )}{2 π} + 1 : \frac{3 arccos ( \frac{5}{12} ) + 2 π}{2 π}

\frac{3 arccos ( \frac{5}{12} )}{2 π} + 1

Convert element to fraction:

1 = \frac{1 \cdot 2 π}{2 π}

= \frac{3 arccos ( \frac{5}{12} )}{2 π} + \frac{1 \cdot 2 π}{2 π}

Since the denominators are equal, combine the fractions:

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

= \frac{3 arccos ( \frac{5}{12} ) + 1 \cdot 2 π}{2 π}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{3 arccos ( \frac{5}{12} ) + 2 π}{2 π}

t > \frac{3 arccos ( \frac{5}{12} ) + 2 π}{2 π} + 3 n

\frac{2 π}{3} (t - 1) < 2 π - arccos (\frac{5}{12}) + 2 πn : t < \frac{8 π - 3 arccos ( \frac{5}{12} )}{2 π} + 3 n

\frac{2 π}{3} (t - 1) < 2 π - arccos (\frac{5}{12}) + 2 πn

Multiply both sides by

3

\frac{2 π}{3} (t - 1) < 2 π - arccos (\frac{5}{12}) + 2 πn

Multiply both sides by

3

3 \cdot \frac{2 π}{3} (t - 1) < 3 \cdot 2 π - 3 arccos (\frac{5}{12}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{2 π}{3} (t - 1) < 3 \cdot 2 π - 3 arccos (\frac{5}{12}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{2 π}{3} (t - 1) : 2 π (t - 1)

3 \cdot \frac{2 π}{3} (t - 1)

Multiply fractions:

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

= \frac{2 \cdot 3 π}{3} (t - 1)

Cancel the common factor:

3

= (t - 1) \cdot 2 π

Simplify

3 \cdot 2 π - 3 arccos (\frac{5}{12}) + 3 \cdot 2 πn : 6 π - 3 arccos (\frac{5}{12}) + 6 πn

3 \cdot 2 π - 3 arccos (\frac{5}{12}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 π - 3 arccos (\frac{5}{12}) + 6 πn

2 π (t - 1) < 6 π - 3 arccos (\frac{5}{12}) + 6 πn

2 π (t - 1) < 6 π - 3 arccos (\frac{5}{12}) + 6 πn

2 π (t - 1) < 6 π - 3 arccos (\frac{5}{12}) + 6 πn

Divide both sides by

2 π

2 π (t - 1) < 6 π - 3 arccos (\frac{5}{12}) + 6 πn

Divide both sides by

2 π

\frac{2 π ( t - 1 )}{2 π} < \frac{6 π}{2 π} - \frac{3 arccos ( \frac{5}{12} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π ( t - 1 )}{2 π} < \frac{6 π}{2 π} - \frac{3 arccos ( \frac{5}{12} )}{2 π} + \frac{6 πn}{2 π}

Simplify

\frac{2 π ( t - 1 )}{2 π} : t - 1

\frac{2 π ( t - 1 )}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π ( t - 1 )}{π}

Cancel the common factor:

π

= t - 1

Simplify

\frac{6 π}{2 π} - \frac{3 arccos ( \frac{5}{12} )}{2 π} + \frac{6 πn}{2 π} : 3 - \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n

\frac{6 π}{2 π} - \frac{3 arccos ( \frac{5}{12} )}{2 π} + \frac{6 πn}{2 π}

Cancel

\frac{6 π}{2 π} : 3

\frac{6 π}{2 π}

Cancel

\frac{6 π}{2 π} : 3

\frac{6 π}{2 π}

Divide the numbers:

\frac{6}{2} = 3

= \frac{3 π}{π}

Cancel the common factor:

π

= 3

= 3

= 3 - \frac{3 arccos ( \frac{5}{12} )}{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

= 3 - \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n

t - 1 < 3 - \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n

t - 1 < 3 - \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n

t - 1 < 3 - \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n

Move

1

to the right side

t - 1 < 3 - \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n

Add

1

to both sides

t - 1 + 1 < 3 - \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n + 1

Simplify

t - 1 + 1 < 3 - \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n + 1

Simplify

t - 1 + 1 : t

t - 1 + 1

Add similar elements:

- 1 + 1 < 0

= t

Simplify

3 - \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n + 1 : 3 n + 4 - \frac{3 arccos ( \frac{5}{12} )}{2 π}

3 - \frac{3 arccos ( \frac{5}{12} )}{2 π} + 3 n + 1

Add the numbers:

3 + 1 = 4

= 3 n + 4 - \frac{3 arccos ( \frac{5}{12} )}{2 π}

t < 3 n + 4 - \frac{3 arccos ( \frac{5}{12} )}{2 π}

t < 3 n + 4 - \frac{3 arccos ( \frac{5}{12} )}{2 π}

t < 3 n + 4 - \frac{3 arccos ( \frac{5}{12} )}{2 π}

Simplify

4 - \frac{3 arccos ( \frac{5}{12} )}{2 π} : \frac{8 π - 3 arccos ( \frac{5}{12} )}{2 π}

4 - \frac{3 arccos ( \frac{5}{12} )}{2 π}

Convert element to fraction:

4 = \frac{4 \cdot 2 π}{2 π}

= \frac{4 \cdot 2 π}{2 π} - \frac{3 arccos ( \frac{5}{12} )}{2 π}

Since the denominators are equal, combine the fractions:

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

= \frac{4 \cdot 2 π - 3 arccos ( \frac{5}{12} )}{2 π}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{8 π - 3 arccos ( \frac{5}{12} )}{2 π}

t < \frac{8 π - 3 arccos ( \frac{5}{12} )}{2 π} + 3 n

Combine the intervals

t > \frac{3 arccos ( \frac{5}{12} ) + 2 π}{2 π} + 3 n and t < \frac{8 π - 3 arccos ( \frac{5}{12} )}{2 π} + 3 n

Merge Overlapping Intervals

\frac{3 arccos ( \frac{5}{12} ) + 2 π}{2 π} + 3 n < t < \frac{8 π - 3 arccos ( \frac{5}{12} )}{2 π} + 3 n

Popular Examples

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