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Popular Trigonometry >

sin(0.5pix)<0.6

Solution

sin (0.5 π x) < 0.6

Solution

\frac{- π - 0.64350 \dots}{1.57079 \dots} + \frac{2}{0.5} n < x < 0.40966 \dots + \frac{2}{0.5} n

Interval Notation

(\frac{- π - 0.64350 \dots}{1.57079 \dots} + \frac{2}{0.5} n, 0.40966 \dots + \frac{2}{0.5} n)

Decimal

- 2.40966 \dots + \frac{2}{0.5} n < x < 0.40966 \dots + \frac{2}{0.5} n

Solution steps

sin (0.5 π x) < 0.6

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (0.6) + 2 πn < 0.5 π x < arcsin (0.6) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (0.6) + 2 πn < 0.5 π x and 0.5 π x < arcsin (0.6) + 2 πn

- π - arcsin (0.6) + 2 πn < 0.5 π x : x > \frac{- π - arcsin ( 0.6 )}{0.5 π} + \frac{2}{0.5} n

- π - arcsin (0.6) + 2 πn < 0.5 π x

Switch sides

0.5 π x > - π - arcsin (0.6) + 2 πn

Divide both sides by

0.5 π

0.5 π x > - π - arcsin (0.6) + 2 πn

Divide both sides by

0.5 π

\frac{0.5 π x}{0.5 π} > - \frac{π}{0.5 π} - \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 πn}{0.5 π}

Simplify

\frac{0.5 π x}{0.5 π} > - \frac{π}{0.5 π} - \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 πn}{0.5 π}

Simplify

\frac{0.5 π x}{0.5 π} : x

\frac{0.5 π x}{0.5 π}

Cancel the common factor:

0.5

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

- \frac{π}{0.5 π} - \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 πn}{0.5 π} : - \frac{1}{0.5} - \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 n}{0.5}

- \frac{π}{0.5 π} - \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 πn}{0.5 π}

Cancel

\frac{π}{0.5 π} : \frac{1}{0.5}

\frac{π}{0.5 π}

Cancel the common factor:

π

= \frac{1}{0.5}

= - \frac{1}{0.5} - \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 πn}{0.5 π}

Cancel

\frac{2 πn}{0.5 π} : \frac{2 n}{0.5}

\frac{2 πn}{0.5 π}

Cancel the common factor:

π

= \frac{2 n}{0.5}

= - \frac{1}{0.5} - \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 n}{0.5}

x > - \frac{1}{0.5} - \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 n}{0.5}

x > - \frac{1}{0.5} - \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 n}{0.5}

Simplify

- \frac{1}{0.5} - \frac{arcsin ( 0.6 )}{0.5 π} : \frac{- π - arcsin ( 0.6 )}{0.5 π}

- \frac{1}{0.5} - \frac{arcsin ( 0.6 )}{0.5 π}

Least Common Multiplier of

0.5, 0.5 π : 0.5 π

0.5, 0.5 π

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

0.5

0.5 π

= 0.5 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

0.5 π

For

\frac{1}{0.5} :

multiply the denominator and numerator by

π

\frac{1}{0.5} = \frac{1 π}{0.5 π} = \frac{π}{0.5 π}

= - \frac{π}{0.5 π} - \frac{arcsin ( 0.6 )}{0.5 π}

Since the denominators are equal, combine the fractions:

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

= \frac{- π - arcsin ( 0.6 )}{0.5 π}

x > \frac{- π - arcsin ( 0.6 )}{0.5 π} + \frac{2}{0.5} n

x > \frac{- π - arcsin ( 0.6 )}{0.5 π} + \frac{2}{0.5} n

0.5 π x < arcsin (0.6) + 2 πn : x < \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 n}{0.5}

0.5 π x < arcsin (0.6) + 2 πn

Divide both sides by

0.5 π

0.5 π x < arcsin (0.6) + 2 πn

Divide both sides by

0.5 π

\frac{0.5 π x}{0.5 π} < \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 πn}{0.5 π}

Simplify

x < \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 n}{0.5}

x < \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 n}{0.5}

Combine the intervals

x > \frac{- π - arcsin ( 0.6 )}{0.5 π} + \frac{2}{0.5} n and x < \frac{arcsin ( 0.6 )}{0.5 π} + \frac{2 n}{0.5}

Merge Overlapping Intervals

\frac{- π - 0.64350 \dots}{1.57079 \dots} + \frac{2}{0.5} n < x < 0.40966 \dots + \frac{2}{0.5} n

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