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0.5sin(x+pi/6)>= (sqrt(3))/4

Solution

0.5 sin (x + \frac{π}{6}) \geq \frac{3}{4}

Solution

\frac{π}{6} + 2 πn \leq x \leq \frac{π}{2} + 2 πn

Interval Notation

[\frac{π}{6} + 2 πn, \frac{π}{2} + 2 πn]

Decimal Notation

0.52359 \dots + 2 πn \leq x \leq 1.57079 \dots + 2 πn

Solution steps

0.5 sin (x + \frac{π}{6}) \geq \frac{3}{4}

Multiply both sides by

10

0.5 sin (x + \frac{π}{6}) \geq \frac{3}{4}

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere is one digit to the right of the decimal point, therefore multiply by

10

0.5 sin (x + \frac{π}{6}) \cdot 10 \geq \frac{3}{4} \cdot 10

Refine

5 sin (x + \frac{π}{6}) \geq \frac{5 3}{2}

5 sin (x + \frac{π}{6}) \geq \frac{5 3}{2}

Divide both sides by

5

5 sin (x + \frac{π}{6}) \geq \frac{5 3}{2}

Divide both sides by

5

\frac{5 sin ( x + \frac{π}{6} )}{5} \geq \frac{\frac{5 3}{2}}{5}

Simplify

\frac{5 sin ( x + \frac{π}{6} )}{5} \geq \frac{\frac{5 3}{2}}{5}

Simplify

\frac{5 sin ( x + \frac{π}{6} )}{5} : sin (x + \frac{π}{6})

\frac{5 sin ( x + \frac{π}{6} )}{5}

Divide the numbers:

\frac{5}{5} = 1

= sin (x + \frac{π}{6})

Simplify

\frac{\frac{5 3}{2}}{5} : \frac{3}{2}

\frac{\frac{5 3}{2}}{5}

Apply the fraction rule:

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

= \frac{5 3}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{5 3}{10}

Cancel the common factor:

5

= \frac{3}{2}

sin (x + \frac{π}{6}) \geq \frac{3}{2}

sin (x + \frac{π}{6}) \geq \frac{3}{2}

sin (x + \frac{π}{6}) \geq \frac{3}{2}

For

sin (x) \geq a

, if

- 1 < a < 1

then

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

arcsin (\frac{3}{2}) + 2 πn \leq (x + \frac{π}{6}) \leq π - arcsin (\frac{3}{2}) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

arcsin (\frac{3}{2}) + 2 πn \leq x + \frac{π}{6} and x + \frac{π}{6} \leq π - arcsin (\frac{3}{2}) + 2 πn

arcsin (\frac{3}{2}) + 2 πn \leq x + \frac{π}{6} : x \geq 2 πn + \frac{π}{6}

arcsin (\frac{3}{2}) + 2 πn \leq x + \frac{π}{6}

Switch sides

x + \frac{π}{6} \geq arcsin (\frac{3}{2}) + 2 πn

Simplify

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

arcsin (\frac{3}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{3} + 2 πn

x + \frac{π}{6} \geq \frac{π}{3} + 2 πn

Move

\frac{π}{6}

to the right side

x + \frac{π}{6} \geq \frac{π}{3} + 2 πn

Subtract

\frac{π}{6}

from both sides

x + \frac{π}{6} - \frac{π}{6} \geq \frac{π}{3} + 2 πn - \frac{π}{6}

Simplify

x + \frac{π}{6} - \frac{π}{6} \geq \frac{π}{3} + 2 πn - \frac{π}{6}

Simplify

x + \frac{π}{6} - \frac{π}{6} : x

x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} \geq 0

= x

Simplify

\frac{π}{3} + 2 πn - \frac{π}{6} : 2 πn + \frac{π}{6}

\frac{π}{3} + 2 πn - \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{3} - \frac{π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

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

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= \frac{π 2}{6} - \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π}{6}

Add similar elements:

2 π - π = π

= 2 πn + \frac{π}{6}

x \geq 2 πn + \frac{π}{6}

x \geq 2 πn + \frac{π}{6}

x \geq 2 πn + \frac{π}{6}

x + \frac{π}{6} \leq π - arcsin (\frac{3}{2}) + 2 πn : x \leq \frac{π}{2} + 2 πn

x + \frac{π}{6} \leq π - arcsin (\frac{3}{2}) + 2 πn

Simplify

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

π - arcsin (\frac{3}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= π - \frac{π}{3} + 2 πn

x + \frac{π}{6} \leq π - \frac{π}{3} + 2 πn

Move

\frac{π}{6}

to the right side

x + \frac{π}{6} \leq π - \frac{π}{3} + 2 πn

Subtract

\frac{π}{6}

from both sides

x + \frac{π}{6} - \frac{π}{6} \leq π - \frac{π}{3} + 2 πn - \frac{π}{6}

Simplify

x + \frac{π}{6} - \frac{π}{6} \leq π - \frac{π}{3} + 2 πn - \frac{π}{6}

Simplify

x + \frac{π}{6} - \frac{π}{6} : x

x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} \leq 0

= x

Simplify

π - \frac{π}{3} + 2 πn - \frac{π}{6} : π + 2 πn - \frac{π}{2}

π - \frac{π}{3} + 2 πn - \frac{π}{6}

Group like terms

= π + 2 πn - \frac{π}{3} - \frac{π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

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

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= - \frac{π 2}{6} - \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 2 - π}{6}

Add similar elements:

- 2 π - π = - 3 π

= \frac{- 3 π}{6}

Apply the fraction rule:

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

= - \frac{3 π}{6}

Cancel the common factor:

3

= π + 2 πn - \frac{π}{2}

x \leq π + 2 πn - \frac{π}{2}

x \leq π + 2 πn - \frac{π}{2}

x \leq π + 2 πn - \frac{π}{2}

Simplify

π - \frac{π}{2} : \frac{π}{2}

π - \frac{π}{2}

Convert element to fraction:

π = \frac{π 2}{2}

= \frac{π 2}{2} - \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π}{2}

Add similar elements:

2 π - π = π

= \frac{π}{2}

x \leq \frac{π}{2} + 2 πn

Combine the intervals

x \geq 2 πn + \frac{π}{6} and x \leq \frac{π}{2} + 2 πn

Merge Overlapping Intervals

\frac{π}{6} + 2 πn \leq x \leq \frac{π}{2} + 2 πn

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