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Popular >

sin((5x)/2)<= 0.6

Solution

sin (\frac{5 x}{2}) \leq 0.6

Solution

- 1.51403 \dots + \frac{4 π}{5} n \leq x \leq 0.25740 \dots + \frac{4 π}{5} n

+1

Interval Notation

[- 1.51403 \dots + \frac{4 π}{5} n, 0.25740 \dots + \frac{4 π}{5} n]

Solution steps

sin (\frac{5 x}{2}) \leq 0.6

For

sin (x) \leq a

, if

- 1 < a < 1

then

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

- π - arcsin (0.6) + 2 πn \leq \frac{5 x}{2} \leq arcsin (0.6) + 2 πn

If

a \leq u \leq b

then

a \leq u and u \leq b

- π - arcsin (0.6) + 2 πn \leq \frac{5 x}{2} and \frac{5 x}{2} \leq arcsin (0.6) + 2 πn

- π - arcsin (0.6) + 2 πn \leq \frac{5 x}{2} : x \geq \frac{- 2 π - 2 arcsin ( 0.6 )}{5} + \frac{4 π}{5} n

- π - arcsin (0.6) + 2 πn \leq \frac{5 x}{2}

Switch sides

\frac{5 x}{2} \geq - π - arcsin (0.6) + 2 πn

Multiply both sides by

2

\frac{5 x}{2} \geq - π - arcsin (0.6) + 2 πn

Multiply both sides by

2

\frac{2 \cdot 5 x}{2} \geq - 2 π - 2 arcsin (0.6) + 2 \cdot 2 πn

Simplify

5 x \geq - 2 π - 2 arcsin (0.6) + 4 πn

5 x \geq - 2 π - 2 arcsin (0.6) + 4 πn

Divide both sides by

5

5 x \geq - 2 π - 2 arcsin (0.6) + 4 πn

Divide both sides by

5

\frac{5 x}{5} \geq - \frac{2 π}{5} - \frac{2 arcsin ( 0.6 )}{5} + \frac{4 πn}{5}

Simplify

x \geq - \frac{2 π}{5} - \frac{2 arcsin ( 0.6 )}{5} + \frac{4 πn}{5}

Simplify

- \frac{2 π}{5} - \frac{2 arcsin ( 0.6 )}{5} : \frac{- 2 π - 2 arcsin ( 0.6 )}{5}

- \frac{2 π}{5} - \frac{2 arcsin ( 0.6 )}{5}

Apply rule

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

= \frac{- 2 π - 2 arcsin ( 0.6 )}{5}

x \geq \frac{- 2 π - 2 arcsin ( 0.6 )}{5} + \frac{4 π}{5} n

x \geq \frac{- 2 π - 2 arcsin ( 0.6 )}{5} + \frac{4 π}{5} n

\frac{5 x}{2} \leq arcsin (0.6) + 2 πn : x \leq \frac{2 arcsin ( 0.6 )}{5} + \frac{4 πn}{5}

\frac{5 x}{2} \leq arcsin (0.6) + 2 πn

Multiply both sides by

2

\frac{5 x}{2} \leq arcsin (0.6) + 2 πn

Multiply both sides by

2

\frac{2 \cdot 5 x}{2} \leq 2 arcsin (0.6) + 2 \cdot 2 πn

Simplify

5 x \leq 2 arcsin (0.6) + 4 πn

5 x \leq 2 arcsin (0.6) + 4 πn

Divide both sides by

5

5 x \leq 2 arcsin (0.6) + 4 πn

Divide both sides by

5

\frac{5 x}{5} \leq \frac{2 arcsin ( 0.6 )}{5} + \frac{4 πn}{5}

Simplify

x \leq \frac{2 arcsin ( 0.6 )}{5} + \frac{4 πn}{5}

x \leq \frac{2 arcsin ( 0.6 )}{5} + \frac{4 πn}{5}

Combine the intervals

x \geq \frac{- 2 π - 2 arcsin ( 0.6 )}{5} + \frac{4 π}{5} n and x \leq \frac{2 arcsin ( 0.6 )}{5} + \frac{4 πn}{5}

Merge Overlapping Intervals

- 1.51403 \dots + \frac{4 π}{5} n \leq x \leq 0.25740 \dots + \frac{4 π}{5} n

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