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Popular Trigonometry >

1/2 (10sin(2pi*60x))<-0.52

Solution

\frac{1}{2} (10 sin (2 π \cdot 60 x)) < - 0.52

Solution

\frac{- π + 0.10418 \dots}{376.99111 \dots} + \frac{1}{60} n < x < - 0.00027 \dots + \frac{1}{60} n

Interval Notation

(\frac{- π + 0.10418 \dots}{376.99111 \dots} + \frac{1}{60} n, - 0.00027 \dots + \frac{1}{60} n)

Decimal

- 0.00805 \dots + \frac{1}{60} n < x < - 0.00027 \dots + \frac{1}{60} n

Solution steps

\frac{1}{2} \cdot 10 sin (2 π 60 x) < - 0.52

Simplify

\frac{1}{2} \cdot 10 : 5

\frac{1}{2} \cdot 10

Convert element to fraction:

10 = \frac{10}{1}

= \frac{1}{2} \cdot \frac{10}{1}

Cross-cancel common factor:

2

= \frac{5}{1}

Apply the fraction rule:

\frac{a}{1} = a

= 5

5 sin (2 π 60 x) < - 0.52

Divide both sides by

5

5 sin (2 π 60 x) < - 0.52

Divide both sides by

5

\frac{5 sin ( 2 π 60 x )}{5} < \frac{- 0.52}{5}

Simplify

\frac{5 sin ( 2 π 60 x )}{5} < \frac{- 0.52}{5}

Simplify

\frac{5 sin ( 2 π 60 x )}{5} : sin (2 π 60 x)

\frac{5 sin ( 2 π 60 x )}{5}

Divide the numbers:

\frac{5}{5} = 1

= sin (2 π 60 x)

Simplify

\frac{- 0.52}{5} : - 0.104

\frac{- 0.52}{5}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{0.52}{5}

Divide the numbers:

\frac{0.52}{5} = 0.104

= - 0.104

sin (2 π 60 x) < - 0.104

sin (2 π 60 x) < - 0.104

sin (2 π 60 x) < - 0.104

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (- 0.104) + 2 πn < 2 π 60 x < arcsin (- 0.104) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (- 0.104) + 2 πn < 2 π 60 x and 2 π 60 x < arcsin (- 0.104) + 2 πn

- π - arcsin (- 0.104) + 2 πn < 2 π 60 x : x > \frac{- π + arcsin ( 0.104 )}{120 π} + \frac{1}{60} n

- π - arcsin (- 0.104) + 2 πn < 2 π 60 x

Switch sides

2 π 60 x > - π - arcsin (- 0.104) + 2 πn

Simplify

- π - arcsin (- 0.104) + 2 πn : - π + arcsin (0.104) + 2 πn

- π - arcsin (- 0.104) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- 0.104) = - arcsin (0.104)

= - π - (- arcsin (0.104)) + 2 πn

Apply rule

- (- a) = a

= - π + arcsin (0.104) + 2 πn

2 π 60 x > - π + arcsin (0.104) + 2 πn

Divide both sides by

120 π

2 π 60 x > - π + arcsin (0.104) + 2 πn

Divide both sides by

120 π

\frac{2 π 60 x}{120 π} > - \frac{π}{120 π} + \frac{arcsin ( 0.104 )}{120 π} + \frac{2 πn}{120 π}

Simplify

\frac{2 π 60 x}{120 π} > - \frac{π}{120 π} + \frac{arcsin ( 0.104 )}{120 π} + \frac{2 πn}{120 π}

Simplify

\frac{2 π 60 x}{120 π} : x

\frac{2 π 60 x}{120 π}

Multiply the numbers:

2 \cdot 60 = 120

= \frac{120 π x}{120 π}

Divide the numbers:

\frac{120}{120} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

- \frac{π}{120 π} + \frac{arcsin ( 0.104 )}{120 π} + \frac{2 πn}{120 π} : - \frac{1}{120} + \frac{arcsin ( 0.104 )}{120 π} + \frac{n}{60}

- \frac{π}{120 π} + \frac{arcsin ( 0.104 )}{120 π} + \frac{2 πn}{120 π}

Cancel

\frac{π}{120 π} : \frac{1}{120}

\frac{π}{120 π}

Cancel the common factor:

π

= \frac{1}{120}

= - \frac{1}{120} + \frac{arcsin ( 0.104 )}{120 π} + \frac{2 πn}{120 π}

Cancel

\frac{2 πn}{120 π} : \frac{n}{60}

\frac{2 πn}{120 π}

Cancel

\frac{2 πn}{120 π} : \frac{n}{60}

\frac{2 πn}{120 π}

Cancel the common factor:

2

= \frac{πn}{60 π}

Cancel the common factor:

π

= \frac{n}{60}

= \frac{n}{60}

= - \frac{1}{120} + \frac{arcsin ( 0.104 )}{120 π} + \frac{n}{60}

x > - \frac{1}{120} + \frac{arcsin ( 0.104 )}{120 π} + \frac{n}{60}

x > - \frac{1}{120} + \frac{arcsin ( 0.104 )}{120 π} + \frac{n}{60}

Simplify

- \frac{1}{120} + \frac{arcsin ( 0.104 )}{120 π} : \frac{- π + arcsin ( 0.104 )}{120 π}

- \frac{1}{120} + \frac{arcsin ( 0.104 )}{120 π}

Least Common Multiplier of

120, 120 π : 120 π

120, 120 π

Lowest Common Multiplier (LCM)

Least Common Multiplier of

120, 120 : 120

120, 120

Least Common Multiplier (LCM)

Prime factorization of

120 : 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

120

120

divides by

2120 = 60 \cdot 2

= 2 \cdot 60

60

divides by

260 = 30 \cdot 2

= 2 \cdot 2 \cdot 30

30

divides by

230 = 15 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

Prime factorization of

120 : 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

120

120

divides by

2120 = 60 \cdot 2

= 2 \cdot 60

60

divides by

260 = 30 \cdot 2

= 2 \cdot 2 \cdot 30

30

divides by

230 = 15 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 15

15

divides by

315 = 5 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

2, 3, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

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

120

120

= 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

Multiply the numbers:

2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 = 120

= 120

Compute an expression comprised of factors that appear either in

120

120 π

= 120 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

120 π

For

\frac{1}{120} :

multiply the denominator and numerator by

π

\frac{1}{120} = \frac{1 π}{120 π} = \frac{π}{120 π}

= - \frac{π}{120 π} + \frac{arcsin ( 0.104 )}{120 π}

Since the denominators are equal, combine the fractions:

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

= \frac{- π + arcsin ( 0.104 )}{120 π}

x > \frac{- π + arcsin ( 0.104 )}{120 π} + \frac{1}{60} n

x > \frac{- π + arcsin ( 0.104 )}{120 π} + \frac{1}{60} n

2 π 60 x < arcsin (- 0.104) + 2 πn : x < - \frac{arcsin ( 0.104 )}{120 π} + \frac{n}{60}

2 π 60 x < arcsin (- 0.104) + 2 πn

Simplify

arcsin (- 0.104) + 2 πn : - arcsin (0.104) + 2 πn

arcsin (- 0.104) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- 0.104) = - arcsin (0.104)

= - arcsin (0.104) + 2 πn

2 π 60 x < - arcsin (0.104) + 2 πn

Divide both sides by

120 π

2 π 60 x < - arcsin (0.104) + 2 πn

Divide both sides by

120 π

\frac{2 π 60 x}{120 π} < - \frac{arcsin ( 0.104 )}{120 π} + \frac{2 πn}{120 π}

Simplify

\frac{2 π 60 x}{120 π} < - \frac{arcsin ( 0.104 )}{120 π} + \frac{2 πn}{120 π}

Simplify

\frac{2 π 60 x}{120 π} : x

\frac{2 π 60 x}{120 π}

Multiply the numbers:

2 \cdot 60 = 120

= \frac{120 π x}{120 π}

Divide the numbers:

\frac{120}{120} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

- \frac{arcsin ( 0.104 )}{120 π} + \frac{2 πn}{120 π} : - \frac{arcsin ( 0.104 )}{120 π} + \frac{n}{60}

- \frac{arcsin ( 0.104 )}{120 π} + \frac{2 πn}{120 π}

Cancel

\frac{2 πn}{120 π} : \frac{n}{60}

\frac{2 πn}{120 π}

Cancel

\frac{2 πn}{120 π} : \frac{n}{60}

\frac{2 πn}{120 π}

Cancel the common factor:

2

= \frac{πn}{60 π}

Cancel the common factor:

π

= \frac{n}{60}

= \frac{n}{60}

= - \frac{arcsin ( 0.104 )}{120 π} + \frac{n}{60}

x < - \frac{arcsin ( 0.104 )}{120 π} + \frac{n}{60}

x < - \frac{arcsin ( 0.104 )}{120 π} + \frac{n}{60}

x < - \frac{arcsin ( 0.104 )}{120 π} + \frac{n}{60}

Combine the intervals

x > \frac{- π + arcsin ( 0.104 )}{120 π} + \frac{1}{60} n and x < - \frac{arcsin ( 0.104 )}{120 π} + \frac{n}{60}

Merge Overlapping Intervals

\frac{- π + 0.10418 \dots}{376.99111 \dots} + \frac{1}{60} n < x < - 0.00027 \dots + \frac{1}{60} n

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