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Popular Trigonometry >

0<82.5-67.5cos(pi/6 t)<20

Solution

0 < 82.5 - 67.5 cos (\frac{π}{6} t) < 20

Solution

- \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n < t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

Interval Notation

(- \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n, \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n)

Decimal

- 0.73972 \dots + 12 n < t < 0.73972 \dots + 12 n

Solution steps

0 < 82.5 - 67.5 cos (\frac{π}{6} t) < 20

a < u < b

then

a < u and u < b

0 < 82.5 - 67.5 cos (\frac{π}{6} t) and 82.5 - 67.5 cos (\frac{π}{6} t) < 20

0 < 82.5 - 67.5 cos (\frac{π}{6} t) :

True for all

t \in R

0 < 82.5 - 67.5 cos (\frac{π}{6} t)

Switch sides

82.5 - 67.5 cos (\frac{π}{6} t) > 0

Multiply both sides by

10

82.5 - 67.5 cos (\frac{π}{6} t) > 0

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

82.5 \cdot 10 - 67.5 cos (\frac{π}{6} t) \cdot 10 > 0 \cdot 10

Refine

825 - 675 cos (\frac{π}{6} t) > 0

825 - 675 cos (\frac{π}{6} t) > 0

Move

825

to the right side

825 - 675 cos (\frac{π}{6} t) > 0

Subtract

825

from both sides

825 - 675 cos (\frac{π}{6} t) - 825 > 0 - 825

Simplify

- 675 cos (\frac{π}{6} t) > - 825

- 675 cos (\frac{π}{6} t) > - 825

Multiply both sides by

- 1

- 675 cos (\frac{π}{6} t) > - 825

Multiply both sides by -1 (reverse the inequality)

(- 675 cos (\frac{π}{6} t)) (- 1) < (- 825) (- 1)

Simplify

675 cos (\frac{π}{6} t) < 825

675 cos (\frac{π}{6} t) < 825

Divide both sides by

675

675 cos (\frac{π}{6} t) < 825

Divide both sides by

675

\frac{675 cos ( \frac{π}{6} t )}{675} < \frac{825}{675}

Simplify

cos (\frac{π}{6} t) < \frac{11}{9}

cos (\frac{π}{6} t) < \frac{11}{9}

Range of

cos (\frac{π}{6} t) : - 1 \leq cos (\frac{π}{6} t) \leq 1

Function range definition

The range of the basic

cos

function is

- 1 \leq cos (\frac{π}{6} t) \leq 1

- 1 \leq cos (\frac{π}{6} t) \leq 1

cos (\frac{π}{6} t) < \frac{11}{9} and - 1 \leq cos (\frac{π}{6} t) \leq 1 : - 1 \leq cos (\frac{π}{6} t) \leq 1

Let

y = cos (\frac{π}{6} t)

Combine the intervals

y < \frac{11}{9} and - 1 \leq y \leq 1

Merge Overlapping Intervals

y < \frac{11}{9} and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y < \frac{11}{9}

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all t

True for all t \in R

82.5 - 67.5 cos (\frac{π}{6} t) < 20 : - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n < t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

82.5 - 67.5 cos (\frac{π}{6} t) < 20

Multiply both sides by

10

82.5 - 67.5 cos (\frac{π}{6} t) < 20

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

82.5 \cdot 10 - 67.5 cos (\frac{π}{6} t) \cdot 10 < 20 \cdot 10

Refine

825 - 675 cos (\frac{π}{6} t) < 200

825 - 675 cos (\frac{π}{6} t) < 200

Move

825

to the right side

825 - 675 cos (\frac{π}{6} t) < 200

Subtract

825

from both sides

825 - 675 cos (\frac{π}{6} t) - 825 < 200 - 825

Simplify

- 675 cos (\frac{π}{6} t) < - 625

- 675 cos (\frac{π}{6} t) < - 625

Multiply both sides by

- 1

- 675 cos (\frac{π}{6} t) < - 625

Multiply both sides by -1 (reverse the inequality)

(- 675 cos (\frac{π}{6} t)) (- 1) > (- 625) (- 1)

Simplify

675 cos (\frac{π}{6} t) > 625

675 cos (\frac{π}{6} t) > 625

Divide both sides by

675

675 cos (\frac{π}{6} t) > 625

Divide both sides by

675

\frac{675 cos ( \frac{π}{6} t )}{675} > \frac{625}{675}

Simplify

cos (\frac{π}{6} t) > \frac{25}{27}

cos (\frac{π}{6} t) > \frac{25}{27}

For

cos (x) > a

, if

- 1 \leq a < 1

then

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

- arccos (\frac{25}{27}) + 2 πn < \frac{π}{6} t < arccos (\frac{25}{27}) + 2 πn

a < u < b

then

a < u and u < b

- arccos (\frac{25}{27}) + 2 πn < \frac{π}{6} t and \frac{π}{6} t < arccos (\frac{25}{27}) + 2 πn

- arccos (\frac{25}{27}) + 2 πn < \frac{π}{6} t : t > - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

- arccos (\frac{25}{27}) + 2 πn < \frac{π}{6} t

Switch sides

\frac{π}{6} t > - arccos (\frac{25}{27}) + 2 πn

Multiply both sides by

6

\frac{π}{6} t > - arccos (\frac{25}{27}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} t > - 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} t > - 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} t : π t

6 \cdot \frac{π}{6} t

Multiply fractions:

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

= \frac{6 π}{6} t

Cancel the common factor:

6

= t π

Simplify

- 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn : - 6 arccos (\frac{25}{27}) + 12 πn

- 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= - 6 arccos (\frac{25}{27}) + 12 πn

π t > - 6 arccos (\frac{25}{27}) + 12 πn

π t > - 6 arccos (\frac{25}{27}) + 12 πn

π t > - 6 arccos (\frac{25}{27}) + 12 πn

Divide both sides by

π

π t > - 6 arccos (\frac{25}{27}) + 12 πn

Divide both sides by

π

\frac{π t}{π} > - \frac{6 arccos ( \frac{25}{27} )}{π} + \frac{12 πn}{π}

Simplify

t > - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

t > - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

\frac{π}{6} t < arccos (\frac{25}{27}) + 2 πn : t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

\frac{π}{6} t < arccos (\frac{25}{27}) + 2 πn

Multiply both sides by

6

\frac{π}{6} t < arccos (\frac{25}{27}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} t < 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} t < 6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} t : π t

6 \cdot \frac{π}{6} t

Multiply fractions:

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

= \frac{6 π}{6} t

Cancel the common factor:

6

= t π

Simplify

6 arccos (\frac{25}{27}) + 6 \cdot 2 πn : 6 arccos (\frac{25}{27}) + 12 πn

6 arccos (\frac{25}{27}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 6 arccos (\frac{25}{27}) + 12 πn

π t < 6 arccos (\frac{25}{27}) + 12 πn

π t < 6 arccos (\frac{25}{27}) + 12 πn

π t < 6 arccos (\frac{25}{27}) + 12 πn

Divide both sides by

π

π t < 6 arccos (\frac{25}{27}) + 12 πn

Divide both sides by

π

\frac{π t}{π} < \frac{6 arccos ( \frac{25}{27} )}{π} + \frac{12 πn}{π}

Simplify

t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

Combine the intervals

t > - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n and t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

Merge Overlapping Intervals

- \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n < t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

Combine the intervals

True for all t \in R and - \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n < t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

Merge Overlapping Intervals

- \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n < t < \frac{6 arccos ( \frac{25}{27} )}{π} + 12 n

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