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50sin(-pi/2 x-pi/2)>= 15

Solution

50 sin (- \frac{π}{2} x - \frac{π}{2}) \geq 15

Solution

\frac{- 3 π + 2 arcsin ( \frac{3}{10} )}{π} + 4 n \leq x \leq \frac{- 2 arcsin ( \frac{3}{10} ) - π}{π} + 4 n

Interval Notation

[\frac{- 3 π + 2 arcsin ( \frac{3}{10} )}{π} + 4 n, \frac{- 2 arcsin ( \frac{3}{10} ) - π}{π} + 4 n]

Decimal Notation

- 2.80602 \dots + 4 n \leq x \leq - 1.19397 \dots + 4 n

Solution steps

50 sin (- \frac{π}{2} x - \frac{π}{2}) \geq 15

Divide both sides by

50

50 sin (- \frac{π}{2} x - \frac{π}{2}) \geq 15

Divide both sides by

50

\frac{50 sin ( - \frac{π}{2} x - \frac{π}{2} )}{50} \geq \frac{15}{50}

Simplify

sin (- \frac{π}{2} x - \frac{π}{2}) \geq \frac{3}{10}

sin (- \frac{π}{2} x - \frac{π}{2}) \geq \frac{3}{10}

Factor out

- 1

from

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

sin (- (\frac{π}{2} x + \frac{π}{2})) \geq \frac{3}{10}

Use the following identity:

sin (- x) = - sin (x)

- sin (\frac{π}{2} + x \frac{π}{2}) \geq \frac{3}{10}

Multiply both sides by

- 1

- sin (\frac{π}{2} + x \frac{π}{2}) \geq \frac{3}{10}

Multiply both sides by -1 (reverse the inequality)

(- sin (\frac{π}{2} + x \frac{π}{2})) (- 1) \leq \frac{3 ( - 1 )}{10}

Simplify

sin (\frac{π}{2} + x \frac{π}{2}) \leq - \frac{3}{10}

sin (\frac{π}{2} + x \frac{π}{2}) \leq - \frac{3}{10}

For

sin (x) \leq a

, if

- 1 < a < 1

then

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

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

a \leq u \leq b

then

a \leq u and u \leq b

- π - arcsin (- \frac{3}{10}) + 2 πn \leq \frac{π}{2} + x \frac{π}{2} and \frac{π}{2} + x \frac{π}{2} \leq arcsin (- \frac{3}{10}) + 2 πn

- π - arcsin (- \frac{3}{10}) + 2 πn \leq \frac{π}{2} + x \frac{π}{2} : x \geq \frac{- 3 π + 2 arcsin ( \frac{3}{10} )}{π} + 4 n

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

Switch sides

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

Simplify

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

- π - arcsin (- \frac{3}{10}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{10}) = - arcsin (\frac{3}{10})

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

Apply rule

- (- a) = a

= - π + arcsin (\frac{3}{10}) + 2 πn

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

Move

\frac{π}{2}

to the right side

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

Subtract

\frac{π}{2}

from both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

2 x \frac{π}{2} : π x

2 x \frac{π}{2}

Multiply fractions:

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

= \frac{2 π}{2} x

Cancel the common factor:

2

= x π

Simplify

- 2 π + 2 arcsin (\frac{3}{10}) + 2 \cdot 2 πn - 2 \cdot \frac{π}{2} : - 3 π + 4 πn + 2 arcsin (\frac{3}{10})

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

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

= - 2 π + 2 arcsin (\frac{3}{10}) + 4 πn - π

Group like terms

= - 2 π - π + 4 πn + 2 arcsin (\frac{3}{10})

Add similar elements:

- 2 π - π = - 3 π

= - 3 π + 4 πn + 2 arcsin (\frac{3}{10})

π x \geq - 3 π + 4 πn + 2 arcsin (\frac{3}{10})

π x \geq - 3 π + 4 πn + 2 arcsin (\frac{3}{10})

π x \geq - 3 π + 4 πn + 2 arcsin (\frac{3}{10})

Divide both sides by

π

π x \geq - 3 π + 4 πn + 2 arcsin (\frac{3}{10})

Divide both sides by

π

\frac{π x}{π} \geq - \frac{3 π}{π} + \frac{4 πn}{π} + \frac{2 arcsin ( \frac{3}{10} )}{π}

Simplify

\frac{π x}{π} \geq - \frac{3 π}{π} + \frac{4 πn}{π} + \frac{2 arcsin ( \frac{3}{10} )}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

- \frac{3 π}{π} + \frac{4 πn}{π} + \frac{2 arcsin ( \frac{3}{10} )}{π} : - 3 + 4 n + \frac{2 arcsin ( \frac{3}{10} )}{π}

- \frac{3 π}{π} + \frac{4 πn}{π} + \frac{2 arcsin ( \frac{3}{10} )}{π}

Cancel

\frac{3 π}{π} : 3

\frac{3 π}{π}

Cancel the common factor:

π

= 3

= - 3 + \frac{4 πn}{π} + \frac{2 arcsin ( \frac{3}{10} )}{π}

Cancel

\frac{4 πn}{π} : 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= - 3 + 4 n + \frac{2 arcsin ( \frac{3}{10} )}{π}

x \geq - 3 + 4 n + \frac{2 arcsin ( \frac{3}{10} )}{π}

x \geq - 3 + 4 n + \frac{2 arcsin ( \frac{3}{10} )}{π}

Simplify

- 3 + \frac{2 arcsin ( \frac{3}{10} )}{π} : \frac{- 3 π + 2 arcsin ( \frac{3}{10} )}{π}

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

Convert element to fraction:

3 = \frac{3 π}{π}

= - \frac{3 π}{π} + \frac{2 arcsin ( \frac{3}{10} )}{π}

Since the denominators are equal, combine the fractions:

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

= \frac{- 3 π + 2 arcsin ( \frac{3}{10} )}{π}

x \geq \frac{- 3 π + 2 arcsin ( \frac{3}{10} )}{π} + 4 n

x \geq \frac{- 3 π + 2 arcsin ( \frac{3}{10} )}{π} + 4 n

\frac{π}{2} + x \frac{π}{2} \leq arcsin (- \frac{3}{10}) + 2 πn : x \leq \frac{- 2 arcsin ( \frac{3}{10} ) - π}{π} + 4 n

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

Simplify

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

arcsin (- \frac{3}{10}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{10}) = - arcsin (\frac{3}{10})

= - arcsin (\frac{3}{10}) + 2 πn

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

Move

\frac{π}{2}

to the right side

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

Subtract

\frac{π}{2}

from both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

2 x \frac{π}{2} \leq - 2 arcsin (\frac{3}{10}) + 2 \cdot 2 πn - 2 \cdot \frac{π}{2}

Simplify

2 x \frac{π}{2} \leq - 2 arcsin (\frac{3}{10}) + 2 \cdot 2 πn - 2 \cdot \frac{π}{2}

Simplify

2 x \frac{π}{2} : π x

2 x \frac{π}{2}

Multiply fractions:

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

= \frac{2 π}{2} x

Cancel the common factor:

2

= x π

Simplify

- 2 arcsin (\frac{3}{10}) + 2 \cdot 2 πn - 2 \cdot \frac{π}{2} : - 2 arcsin (\frac{3}{10}) + 4 πn - π

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

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

= - 2 arcsin (\frac{3}{10}) + 4 πn - π

π x \leq - 2 arcsin (\frac{3}{10}) + 4 πn - π

π x \leq - 2 arcsin (\frac{3}{10}) + 4 πn - π

π x \leq - 2 arcsin (\frac{3}{10}) + 4 πn - π

Divide both sides by

π

π x \leq - 2 arcsin (\frac{3}{10}) + 4 πn - π

Divide both sides by

π

\frac{π x}{π} \leq - \frac{2 arcsin ( \frac{3}{10} )}{π} + \frac{4 πn}{π} - \frac{π}{π}

Simplify

\frac{π x}{π} \leq - \frac{2 arcsin ( \frac{3}{10} )}{π} + \frac{4 πn}{π} - \frac{π}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

- \frac{2 arcsin ( \frac{3}{10} )}{π} + \frac{4 πn}{π} - \frac{π}{π} : - \frac{2 arcsin ( \frac{3}{10} )}{π} + 4 n - 1

- \frac{2 arcsin ( \frac{3}{10} )}{π} + \frac{4 πn}{π} - \frac{π}{π}

Apply rule

\frac{a}{a} = 1

\frac{π}{π} = 1

= - \frac{2 arcsin ( \frac{3}{10} )}{π} + \frac{4 πn}{π} - 1

Cancel

\frac{4 πn}{π} : 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= - \frac{2 arcsin ( \frac{3}{10} )}{π} + 4 n - 1

x \leq - \frac{2 arcsin ( \frac{3}{10} )}{π} + 4 n - 1

x \leq - \frac{2 arcsin ( \frac{3}{10} )}{π} + 4 n - 1

Simplify

- \frac{2 arcsin ( \frac{3}{10} )}{π} - 1 : \frac{- 2 arcsin ( \frac{3}{10} ) - π}{π}

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

Convert element to fraction:

1 = \frac{1 π}{π}

= - \frac{2 arcsin ( \frac{3}{10} )}{π} - \frac{1 π}{π}

Since the denominators are equal, combine the fractions:

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

= \frac{- 2 arcsin ( \frac{3}{10} ) - 1 π}{π}

Multiply:

1 π = π

= \frac{- 2 arcsin ( \frac{3}{10} ) - π}{π}

x \leq \frac{- 2 arcsin ( \frac{3}{10} ) - π}{π} + 4 n

x \leq \frac{- 2 arcsin ( \frac{3}{10} ) - π}{π} + 4 n

Combine the intervals

x \geq \frac{- 3 π + 2 arcsin ( \frac{3}{10} )}{π} + 4 n and x \leq \frac{- 2 arcsin ( \frac{3}{10} ) - π}{π} + 4 n

Merge Overlapping Intervals

\frac{- 3 π + 2 arcsin ( \frac{3}{10} )}{π} + 4 n \leq x \leq \frac{- 2 arcsin ( \frac{3}{10} ) - π}{π} + 4 n

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