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Popular Trigonometry >

-1/5 sin((2pi)/5 (x+1))+1<= 16/15

Solution

- \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) + 1 \leq \frac{16}{15}

Solution

\frac{- 5 arcsin ( \frac{1}{3} ) - 2 π}{2 π} + 5 n \leq x \leq \frac{3 π + 5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

Interval Notation

[\frac{- 5 arcsin ( \frac{1}{3} ) - 2 π}{2 π} + 5 n, \frac{3 π + 5 arcsin ( \frac{1}{3} )}{2 π} + 5 n]

Decimal

- 1.27043 \dots + 5 n \leq x \leq 1.77043 \dots + 5 n

Solution steps

- \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) + 1 \leq \frac{16}{15}

Move

1

to the right side

- \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) + 1 \leq \frac{16}{15}

Subtract

1

from both sides

- \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) + 1 - 1 \leq \frac{16}{15} - 1

Simplify

- \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) + 1 - 1 \leq \frac{16}{15} - 1

Simplify

- \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) + 1 - 1 : - \frac{1}{5} sin (\frac{2 π}{5} (x + 1))

- \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) + 1 - 1

Add similar elements:

1 - 1 \leq 0

= - \frac{1}{5} sin (\frac{2 π}{5} (x + 1))

Simplify

\frac{16}{15} - 1 : \frac{1}{15}

\frac{16}{15} - 1

Convert element to fraction:

1 = \frac{1 \cdot 15}{15}

= - \frac{1 \cdot 15}{15} + \frac{16}{15}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 15 + 16}{15}

- 1 \cdot 15 + 16 = 1

- 1 \cdot 15 + 16

Multiply the numbers:

1 \cdot 15 = 15

= - 15 + 16

Add/Subtract the numbers:

- 15 + 16 = 1

= 1

= \frac{1}{15}

- \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) \leq \frac{1}{15}

- \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) \leq \frac{1}{15}

- \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) \leq \frac{1}{15}

Multiply both sides by

- 1

- \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) \leq \frac{1}{15}

Multiply both sides by -1 (reverse the inequality)

(- \frac{1}{5} sin (\frac{2 π}{5} (x + 1))) (- 1) \geq \frac{1 \cdot ( - 1 )}{15}

Simplify

\frac{1}{5} sin (\frac{2 π}{5} (x + 1)) \geq - \frac{1}{15}

\frac{1}{5} sin (\frac{2 π}{5} (x + 1)) \geq - \frac{1}{15}

Multiply both sides by

5

\frac{1}{5} sin (\frac{2 π}{5} (x + 1)) \geq - \frac{1}{15}

Multiply both sides by

5

5 \cdot \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) \geq 5 (- \frac{1}{15})

Simplify

5 \cdot \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) \geq 5 (- \frac{1}{15})

Simplify

5 \cdot \frac{1}{5} sin (\frac{2 π}{5} (x + 1)) : sin (\frac{2 π}{5} (x + 1))

5 \cdot \frac{1}{5} sin (\frac{2 π}{5} (x + 1))

Multiply fractions:

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

= \frac{1 \cdot 5}{5} sin (\frac{2 π}{5} (x + 1))

Cancel the common factor:

5

= sin (\frac{2 π}{5} (x + 1)) \cdot 1

Multiply:

sin (\frac{2 π}{5} (x + 1)) \cdot 1 = sin (\frac{2 π}{5} (x + 1))

= sin (\frac{2 π}{5} (x + 1))

Simplify

5 (- \frac{1}{15}) : - \frac{1}{3}

5 (- \frac{1}{15})

Remove parentheses:

(- a) = - a

= - 5 \cdot \frac{1}{15}

Multiply fractions:

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

= - \frac{1 \cdot 5}{15}

Multiply the numbers:

1 \cdot 5 = 5

= - \frac{5}{15}

Cancel the common factor:

5

= - \frac{1}{3}

sin (\frac{2 π}{5} (x + 1)) \geq - \frac{1}{3}

sin (\frac{2 π}{5} (x + 1)) \geq - \frac{1}{3}

sin (\frac{2 π}{5} (x + 1)) \geq - \frac{1}{3}

For

sin (x) \geq a

, if

- 1 < a < 1

then

arcsin (a) + 2 πn \leq x \leq π - arcsin (a) + 2 πn

arcsin (- \frac{1}{3}) + 2 πn \leq \frac{2 π}{5} (x + 1) \leq π - arcsin (- \frac{1}{3}) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

arcsin (- \frac{1}{3}) + 2 πn \leq \frac{2 π}{5} (x + 1) and \frac{2 π}{5} (x + 1) \leq π - arcsin (- \frac{1}{3}) + 2 πn

arcsin (- \frac{1}{3}) + 2 πn \leq \frac{2 π}{5} (x + 1) : x \geq \frac{- 5 arcsin ( \frac{1}{3} ) - 2 π}{2 π} + 5 n

arcsin (- \frac{1}{3}) + 2 πn \leq \frac{2 π}{5} (x + 1)

Switch sides

\frac{2 π}{5} (x + 1) \geq arcsin (- \frac{1}{3}) + 2 πn

Simplify

arcsin (- \frac{1}{3}) + 2 πn : - arcsin (\frac{1}{3}) + 2 πn

arcsin (- \frac{1}{3}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{3}) = - arcsin (\frac{1}{3})

= - arcsin (\frac{1}{3}) + 2 πn

\frac{2 π}{5} (x + 1) \geq - arcsin (\frac{1}{3}) + 2 πn

Multiply both sides by

5

\frac{2 π}{5} (x + 1) \geq - arcsin (\frac{1}{3}) + 2 πn

Multiply both sides by

5

5 \cdot \frac{2 π}{5} (x + 1) \geq - 5 arcsin (\frac{1}{3}) + 5 \cdot 2 πn

Simplify

5 \cdot \frac{2 π}{5} (x + 1) \geq - 5 arcsin (\frac{1}{3}) + 5 \cdot 2 πn

Simplify

5 \cdot \frac{2 π}{5} (x + 1) : 2 π (x + 1)

5 \cdot \frac{2 π}{5} (x + 1)

Multiply fractions:

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

= \frac{2 \cdot 5 π}{5} (x + 1)

Cancel the common factor:

5

= (x + 1) \cdot 2 π

Simplify

- 5 arcsin (\frac{1}{3}) + 5 \cdot 2 πn : - 5 arcsin (\frac{1}{3}) + 10 πn

- 5 arcsin (\frac{1}{3}) + 5 \cdot 2 πn

Multiply the numbers:

5 \cdot 2 = 10

= - 5 arcsin (\frac{1}{3}) + 10 πn

2 π (x + 1) \geq - 5 arcsin (\frac{1}{3}) + 10 πn

2 π (x + 1) \geq - 5 arcsin (\frac{1}{3}) + 10 πn

2 π (x + 1) \geq - 5 arcsin (\frac{1}{3}) + 10 πn

Divide both sides by

2 π

2 π (x + 1) \geq - 5 arcsin (\frac{1}{3}) + 10 πn

Divide both sides by

2 π

\frac{2 π ( x + 1 )}{2 π} \geq - \frac{5 arcsin ( \frac{1}{3} )}{2 π} + \frac{10 πn}{2 π}

Simplify

\frac{2 π ( x + 1 )}{2 π} \geq - \frac{5 arcsin ( \frac{1}{3} )}{2 π} + \frac{10 πn}{2 π}

Simplify

\frac{2 π ( x + 1 )}{2 π} : x + 1

\frac{2 π ( x + 1 )}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π ( x + 1 )}{π}

Cancel the common factor:

π

= x + 1

Simplify

- \frac{5 arcsin ( \frac{1}{3} )}{2 π} + \frac{10 πn}{2 π} : - \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

- \frac{5 arcsin ( \frac{1}{3} )}{2 π} + \frac{10 πn}{2 π}

Cancel

\frac{10 πn}{2 π} : 5 n

\frac{10 πn}{2 π}

Cancel

\frac{10 πn}{2 π} : 5 n

\frac{10 πn}{2 π}

Divide the numbers:

\frac{10}{2} = 5

= \frac{5 πn}{π}

Cancel the common factor:

π

= 5 n

= 5 n

= - \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

x + 1 \geq - \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

x + 1 \geq - \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

x + 1 \geq - \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

Move

1

to the right side

x + 1 \geq - \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

Subtract

1

from both sides

x + 1 - 1 \geq - \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n - 1

Simplify

x \geq - \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n - 1

x \geq - \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n - 1

Simplify

- \frac{5 arcsin ( \frac{1}{3} )}{2 π} - 1 : \frac{- 5 arcsin ( \frac{1}{3} ) - 2 π}{2 π}

- \frac{5 arcsin ( \frac{1}{3} )}{2 π} - 1

Convert element to fraction:

1 = \frac{1 \cdot 2 π}{2 π}

= - \frac{5 arcsin ( \frac{1}{3} )}{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{- 5 arcsin ( \frac{1}{3} ) - 1 \cdot 2 π}{2 π}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{- 5 arcsin ( \frac{1}{3} ) - 2 π}{2 π}

x \geq \frac{- 5 arcsin ( \frac{1}{3} ) - 2 π}{2 π} + 5 n

\frac{2 π}{5} (x + 1) \leq π - arcsin (- \frac{1}{3}) + 2 πn : x \leq \frac{3 π + 5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

\frac{2 π}{5} (x + 1) \leq π - arcsin (- \frac{1}{3}) + 2 πn

Simplify

π - arcsin (- \frac{1}{3}) + 2 πn : π + arcsin (\frac{1}{3}) + 2 πn

π - arcsin (- \frac{1}{3}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{3}) = - arcsin (\frac{1}{3})

= π - (- arcsin (\frac{1}{3})) + 2 πn

Apply rule

- (- a) = a

= π + arcsin (\frac{1}{3}) + 2 πn

\frac{2 π}{5} (x + 1) \leq π + arcsin (\frac{1}{3}) + 2 πn

Multiply both sides by

5

\frac{2 π}{5} (x + 1) \leq π + arcsin (\frac{1}{3}) + 2 πn

Multiply both sides by

5

5 \cdot \frac{2 π}{5} (x + 1) \leq 5 π + 5 arcsin (\frac{1}{3}) + 5 \cdot 2 πn

Simplify

5 \cdot \frac{2 π}{5} (x + 1) \leq 5 π + 5 arcsin (\frac{1}{3}) + 5 \cdot 2 πn

Simplify

5 \cdot \frac{2 π}{5} (x + 1) : 2 π (x + 1)

5 \cdot \frac{2 π}{5} (x + 1)

Multiply fractions:

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

= \frac{2 \cdot 5 π}{5} (x + 1)

Cancel the common factor:

5

= (x + 1) \cdot 2 π

Simplify

5 π + 5 arcsin (\frac{1}{3}) + 5 \cdot 2 πn : 5 π + 5 arcsin (\frac{1}{3}) + 10 πn

5 π + 5 arcsin (\frac{1}{3}) + 5 \cdot 2 πn

Multiply the numbers:

5 \cdot 2 = 10

= 5 π + 5 arcsin (\frac{1}{3}) + 10 πn

2 π (x + 1) \leq 5 π + 5 arcsin (\frac{1}{3}) + 10 πn

2 π (x + 1) \leq 5 π + 5 arcsin (\frac{1}{3}) + 10 πn

2 π (x + 1) \leq 5 π + 5 arcsin (\frac{1}{3}) + 10 πn

Divide both sides by

2 π

2 π (x + 1) \leq 5 π + 5 arcsin (\frac{1}{3}) + 10 πn

Divide both sides by

2 π

\frac{2 π ( x + 1 )}{2 π} \leq \frac{5 π}{2 π} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + \frac{10 πn}{2 π}

Simplify

\frac{2 π ( x + 1 )}{2 π} \leq \frac{5 π}{2 π} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + \frac{10 πn}{2 π}

Simplify

\frac{2 π ( x + 1 )}{2 π} : x + 1

\frac{2 π ( x + 1 )}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π ( x + 1 )}{π}

Cancel the common factor:

π

= x + 1

Simplify

\frac{5 π}{2 π} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + \frac{10 πn}{2 π} : \frac{5}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

\frac{5 π}{2 π} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + \frac{10 πn}{2 π}

Cancel

\frac{5 π}{2 π} : \frac{5}{2}

\frac{5 π}{2 π}

Cancel the common factor:

π

= \frac{5}{2}

= \frac{5}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + \frac{10 πn}{2 π}

Cancel

\frac{10 πn}{2 π} : 5 n

\frac{10 πn}{2 π}

Cancel

\frac{10 πn}{2 π} : 5 n

\frac{10 πn}{2 π}

Divide the numbers:

\frac{10}{2} = 5

= \frac{5 πn}{π}

Cancel the common factor:

π

= 5 n

= 5 n

= \frac{5}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

x + 1 \leq \frac{5}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

x + 1 \leq \frac{5}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

x + 1 \leq \frac{5}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

Move

1

to the right side

x + 1 \leq \frac{5}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

Subtract

1

from both sides

x + 1 - 1 \leq \frac{5}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n - 1

Simplify

x + 1 - 1 \leq \frac{5}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n - 1

Simplify

x + 1 - 1 : x

x + 1 - 1

Add similar elements:

1 - 1 \leq 0

= x

Simplify

\frac{5}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π} + 5 n - 1 : 5 n + \frac{3}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π}

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

Combine the fractions

- 1 + \frac{5}{2} : \frac{3}{2}

- 1 + \frac{5}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= - \frac{1 \cdot 2}{2} + \frac{5}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot 2 + 5}{2}

- 1 \cdot 2 + 5 = 3

- 1 \cdot 2 + 5

Multiply the numbers:

1 \cdot 2 = 2

= - 2 + 5

Add/Subtract the numbers:

- 2 + 5 = 3

= 3

= \frac{3}{2}

= 5 n + \frac{3}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π}

x \leq 5 n + \frac{3}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π}

x \leq 5 n + \frac{3}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π}

x \leq 5 n + \frac{3}{2} + \frac{5 arcsin ( \frac{1}{3} )}{2 π}

Simplify

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

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

Least Common Multiplier of

2, 2 π : 2 π

2, 2 π

Lowest Common Multiplier (LCM)

Least Common Multiplier of

2, 2 : 2

2, 2

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

2

2

= 2

Multiply the numbers:

2 = 2

= 2

Compute an expression comprised of factors that appear either in

2

2 π

= 2 π

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

2 π

For

\frac{3}{2} :

multiply the denominator and numerator by

π

\frac{3}{2} = \frac{3 π}{2 π}

= \frac{3 π}{2 π} + \frac{5 arcsin ( \frac{1}{3} )}{2 π}

Since the denominators are equal, combine the fractions:

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

= \frac{3 π + 5 arcsin ( \frac{1}{3} )}{2 π}

x \leq \frac{3 π + 5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

Combine the intervals

x \geq \frac{- 5 arcsin ( \frac{1}{3} ) - 2 π}{2 π} + 5 n and x \leq \frac{3 π + 5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

Merge Overlapping Intervals

\frac{- 5 arcsin ( \frac{1}{3} ) - 2 π}{2 π} + 5 n \leq x \leq \frac{3 π + 5 arcsin ( \frac{1}{3} )}{2 π} + 5 n

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