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Popular Trigonometry >

-6sin(2x-30)>0

Solution

- 6 sin (2 x - 30) > 0

Solution

\frac{- π + 30}{2} + πn < x < 15 + πn

Interval Notation

(\frac{- π + 30}{2} + πn, 15 + πn)

Decimal

13.42920 \dots + πn < x < 15 + πn

Solution steps

- 6 sin (2 x - 30) > 0

Multiply both sides by

- 1

- 6 sin (2 x - 30) > 0

Multiply both sides by -1 (reverse the inequality)

(- 6 sin (2 x - 30)) (- 1) < 0 \cdot (- 1)

Simplify

6 sin (2 x - 30) < 0

6 sin (2 x - 30) < 0

Divide both sides by

6

6 sin (2 x - 30) < 0

Divide both sides by

6

\frac{6 sin ( 2 x - 30 )}{6} < \frac{0}{6}

Simplify

sin (2 x - 30) < 0

sin (2 x - 30) < 0

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (0) + 2 πn < (2 x - 30) < arcsin (0) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (0) + 2 πn < 2 x - 30 and 2 x - 30 < arcsin (0) + 2 πn

- π - arcsin (0) + 2 πn < 2 x - 30 : x > \frac{- π + 30}{2} + πn

- π - arcsin (0) + 2 πn < 2 x - 30

Switch sides

2 x - 30 > - π - arcsin (0) + 2 πn

Simplify

- π - arcsin (0) + 2 πn : 2 πn - π

- π - arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

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 - 0 + 2 πn

- π - 0 + 2 πn = - π + 2 πn

= 2 πn - π

2 x - 30 > 2 πn - π

Move

30

to the right side

2 x - 30 > 2 πn - π

Add

30

to both sides

2 x - 30 + 30 > 2 πn - π + 30

Simplify

2 x > 2 πn - π + 30

2 x > 2 πn - π + 30

Divide both sides by

2

2 x > 2 πn - π + 30

Divide both sides by

2

\frac{2 x}{2} > \frac{2 πn}{2} - \frac{π}{2} + \frac{30}{2}

Simplify

\frac{2 x}{2} > \frac{2 πn}{2} - \frac{π}{2} + \frac{30}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} - \frac{π}{2} + \frac{30}{2} : πn - \frac{π}{2} + 15

\frac{2 πn}{2} - \frac{π}{2} + \frac{30}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn - \frac{π}{2} + \frac{30}{2}

Divide the numbers:

\frac{30}{2} = 15

= πn - \frac{π}{2} + 15

x > πn - \frac{π}{2} + 15

x > πn - \frac{π}{2} + 15

Simplify

- \frac{π}{2} + 15 : \frac{- π + 30}{2}

- \frac{π}{2} + 15

Convert element to fraction:

15 = \frac{15 \cdot 2}{2}

= - \frac{π}{2} + \frac{15 \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{- π + 15 \cdot 2}{2}

Multiply the numbers:

15 \cdot 2 = 30

= \frac{- π + 30}{2}

x > \frac{- π + 30}{2} + πn

x > \frac{- π + 30}{2} + πn

2 x - 30 < arcsin (0) + 2 πn : x < πn + 15

2 x - 30 < arcsin (0) + 2 πn

Simplify

arcsin (0) + 2 πn : 2 πn

arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

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}

= 0 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

2 x - 30 < 2 πn

Move

30

to the right side

2 x - 30 < 2 πn

Add

30

to both sides

2 x - 30 + 30 < 2 πn + 30

Simplify

2 x < 2 πn + 30

2 x < 2 πn + 30

Divide both sides by

2

2 x < 2 πn + 30

Divide both sides by

2

\frac{2 x}{2} < \frac{2 πn}{2} + \frac{30}{2}

Simplify

\frac{2 x}{2} < \frac{2 πn}{2} + \frac{30}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} + \frac{30}{2} : πn + 15

\frac{2 πn}{2} + \frac{30}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn + \frac{30}{2}

Divide the numbers:

\frac{30}{2} = 15

= πn + 15

x < πn + 15

x < πn + 15

x < πn + 15

Combine the intervals

x > \frac{- π + 30}{2} + πn and x < πn + 15

Merge Overlapping Intervals

\frac{- π + 30}{2} + πn < x < 15 + πn

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