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Popular >

sin(3X-15)>-(sqrt(2))/2

Solution

sin (3 X - 15) > - \frac{2}{2}

Solution

\frac{60 - π}{12} + \frac{2 π}{3} n < X < \frac{5 π + 60}{12} + \frac{2 π}{3} n

Interval Notation

(\frac{60 - π}{12} + \frac{2 π}{3} n, \frac{5 π + 60}{12} + \frac{2 π}{3} n)

Decimal Notation

4.73820 \dots + \frac{2 π}{3} n < X < 6.30899 \dots + \frac{2 π}{3} n

Solution steps

sin (3 X - 15) > - \frac{2}{2}

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (- \frac{2}{2}) + 2 πn < (3 X - 15) < π - arcsin (- \frac{2}{2}) + 2 πn

a < u < b

then

a < u and u < b

arcsin (- \frac{2}{2}) + 2 πn < 3 X - 15 and 3 X - 15 < π - arcsin (- \frac{2}{2}) + 2 πn

arcsin (- \frac{2}{2}) + 2 πn < 3 X - 15 : X > \frac{60 - π}{12} + \frac{2 π}{3} n

arcsin (- \frac{2}{2}) + 2 πn < 3 X - 15

Switch sides

3 X - 15 > arcsin (- \frac{2}{2}) + 2 πn

Simplify

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

arcsin (- \frac{2}{2}) + 2 πn

arcsin (- \frac{2}{2}) = - \frac{π}{4}

arcsin (- \frac{2}{2})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{2}) = - arcsin (\frac{2}{2})

= - arcsin (\frac{2}{2})

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

arcsin (\frac{2}{2})

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}{4}

= - \frac{π}{4}

= - \frac{π}{4} + 2 πn

3 X - 15 > - \frac{π}{4} + 2 πn

Move

15

to the right side

3 X - 15 > - \frac{π}{4} + 2 πn

Add

15

to both sides

3 X - 15 + 15 > - \frac{π}{4} + 2 πn + 15

Simplify

3 X > - \frac{π}{4} + 2 πn + 15

3 X > - \frac{π}{4} + 2 πn + 15

Divide both sides by

3

3 X > - \frac{π}{4} + 2 πn + 15

Divide both sides by

3

\frac{3 X}{3} > - \frac{\frac{π}{4}}{3} + \frac{2 πn}{3} + \frac{15}{3}

Simplify

\frac{3 X}{3} > - \frac{\frac{π}{4}}{3} + \frac{2 πn}{3} + \frac{15}{3}

Simplify

\frac{3 X}{3} : X

\frac{3 X}{3}

Divide the numbers:

\frac{3}{3} = 1

= X

Simplify

- \frac{\frac{π}{4}}{3} + \frac{2 πn}{3} + \frac{15}{3} : 5 + \frac{2 πn}{3} - \frac{π}{12}

- \frac{\frac{π}{4}}{3} + \frac{2 πn}{3} + \frac{15}{3}

Group like terms

= \frac{15}{3} + \frac{2 πn}{3} - \frac{\frac{π}{4}}{3}

\frac{15}{3} = 5

\frac{15}{3}

Divide the numbers:

\frac{15}{3} = 5

= 5

\frac{\frac{π}{4}}{3} = \frac{π}{12}

\frac{\frac{π}{4}}{3}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{π}{12}

= 5 + \frac{2 πn}{3} - \frac{π}{12}

X > 5 + \frac{2 πn}{3} - \frac{π}{12}

X > 5 + \frac{2 πn}{3} - \frac{π}{12}

Simplify

5 - \frac{π}{12} : \frac{60 - π}{12}

5 - \frac{π}{12}

Convert element to fraction:

5 = \frac{5 \cdot 12}{12}

= \frac{5 \cdot 12}{12} - \frac{π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{5 \cdot 12 - π}{12}

Multiply the numbers:

5 \cdot 12 = 60

= \frac{60 - π}{12}

X > \frac{60 - π}{12} + \frac{2 π}{3} n

X > \frac{60 - π}{12} + \frac{2 π}{3} n

3 X - 15 < π - arcsin (- \frac{2}{2}) + 2 πn : X < \frac{5 π + 60}{12} + \frac{2 π}{3} n

3 X - 15 < π - arcsin (- \frac{2}{2}) + 2 πn

Simplify

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

π - arcsin (- \frac{2}{2}) + 2 πn

arcsin (- \frac{2}{2}) = - \frac{π}{4}

arcsin (- \frac{2}{2})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{2}) = - arcsin (\frac{2}{2})

= - arcsin (\frac{2}{2})

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

arcsin (\frac{2}{2})

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}{4}

= - \frac{π}{4}

= π - (- \frac{π}{4}) + 2 πn

Apply rule

- (- a) = a

= π + \frac{π}{4} + 2 πn

3 X - 15 < π + \frac{π}{4} + 2 πn

Move

15

to the right side

3 X - 15 < π + \frac{π}{4} + 2 πn

Add

15

to both sides

3 X - 15 + 15 < π + \frac{π}{4} + 2 πn + 15

Simplify

3 X < π + \frac{π}{4} + 2 πn + 15

3 X < π + \frac{π}{4} + 2 πn + 15

Divide both sides by

3

3 X < π + \frac{π}{4} + 2 πn + 15

Divide both sides by

3

\frac{3 X}{3} < \frac{π}{3} + \frac{\frac{π}{4}}{3} + \frac{2 πn}{3} + \frac{15}{3}

Simplify

\frac{3 X}{3} < \frac{π}{3} + \frac{\frac{π}{4}}{3} + \frac{2 πn}{3} + \frac{15}{3}

Simplify

\frac{3 X}{3} : X

\frac{3 X}{3}

Divide the numbers:

\frac{3}{3} = 1

= X

Simplify

\frac{π}{3} + \frac{\frac{π}{4}}{3} + \frac{2 πn}{3} + \frac{15}{3} : \frac{π}{3} + \frac{π}{12} + \frac{2 πn}{3} + 5

\frac{π}{3} + \frac{\frac{π}{4}}{3} + \frac{2 πn}{3} + \frac{15}{3}

Group like terms

= \frac{π}{3} + \frac{15}{3} + \frac{2 πn}{3} + \frac{\frac{π}{4}}{3}

\frac{15}{3} = 5

\frac{15}{3}

Divide the numbers:

\frac{15}{3} = 5

= 5

\frac{\frac{π}{4}}{3} = \frac{π}{12}

\frac{\frac{π}{4}}{3}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{π}{12}

= \frac{π}{3} + 5 + \frac{2 πn}{3} + \frac{π}{12}

Group like terms

= \frac{π}{3} + \frac{π}{12} + \frac{2 πn}{3} + 5

X < \frac{π}{3} + \frac{π}{12} + \frac{2 πn}{3} + 5

X < \frac{π}{3} + \frac{π}{12} + \frac{2 πn}{3} + 5

Simplify

\frac{π}{3} + \frac{π}{12} + 5 : \frac{5 π + 60}{12}

\frac{π}{3} + \frac{π}{12} + 5

Convert element to fraction:

5 = \frac{5}{1}

= \frac{π}{3} + \frac{π}{12} + \frac{5}{1}

Least Common Multiplier of

3, 12, 1 : 12

3, 12, 1

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

12 : 2 \cdot 2 \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

Prime factorization of

1

Compute a number comprised of factors that appear in at least one of the following:

3, 12, 1

= 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 2 \cdot 2 = 12

= 12

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

12

For

\frac{π}{3} :

multiply the denominator and numerator by

4

\frac{π}{3} = \frac{π 4}{3 \cdot 4} = \frac{π 4}{12}

For

\frac{5}{1} :

multiply the denominator and numerator by

12

\frac{5}{1} = \frac{5 \cdot 12}{1 \cdot 12} = \frac{60}{12}

= \frac{π 4}{12} + \frac{π}{12} + \frac{60}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 + π + 60}{12}

Add similar elements:

4 π + π = 5 π

= \frac{5 π + 60}{12}

X < \frac{5 π + 60}{12} + \frac{2 π}{3} n

X < \frac{5 π + 60}{12} + \frac{2 π}{3} n

Combine the intervals

X > \frac{60 - π}{12} + \frac{2 π}{3} n and X < \frac{5 π + 60}{12} + \frac{2 π}{3} n

Merge Overlapping Intervals

\frac{60 - π}{12} + \frac{2 π}{3} n < X < \frac{5 π + 60}{12} + \frac{2 π}{3} n

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