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Popular Trigonometry >

solvefor x, 1/(sin(x))-4<0

Solution

solve for

x, \frac{1}{sin ( x )} - 4 < 0

Solution

arcsin (\frac{1}{4}) + 2 πn < x < π - arcsin (\frac{1}{4}) + 2 πn or - π + 2 πn < x < 2 πn

Solution steps

\frac{1}{sin ( x )} - 4 < 0

Rewrite in standard form

\frac{1}{sin ( x )} - 4 < 0

Simplify

\frac{1}{sin ( x )} - 4 : \frac{1 - 4 sin ( x )}{sin ( x )}

\frac{1}{sin ( x )} - 4

Convert element to fraction:

4 = \frac{4 sin ( x )}{sin ( x )}

= \frac{1}{sin ( x )} - \frac{4 sin ( x )}{sin ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 - 4 sin ( x )}{sin ( x )}

\frac{1 - 4 sin ( x )}{sin ( x )} < 0

\frac{1 - 4 sin ( x )}{sin ( x )} < 0

Identify the intervals

Find the signs of the factors of

\frac{1 - 4 sin ( x )}{sin ( x )}

Find the signs of

1 - 4 sin (x)

1 - 4 sin (x) = 0 : sin (x) = \frac{1}{4}

1 - 4 sin (x) = 0

Move

1

to the right side

1 - 4 sin (x) = 0

Subtract

1

from both sides

1 - 4 sin (x) - 1 = 0 - 1

Simplify

- 4 sin (x) = - 1

- 4 sin (x) = - 1

Divide both sides by

- 4

- 4 sin (x) = - 1

Divide both sides by

- 4

\frac{- 4 sin ( x )}{- 4} = \frac{- 1}{- 4}

Simplify

sin (x) = \frac{1}{4}

sin (x) = \frac{1}{4}

1 - 4 sin (x) < 0 : sin (x) > \frac{1}{4}

1 - 4 sin (x) < 0

Move

1

to the right side

1 - 4 sin (x) < 0

Subtract

1

from both sides

1 - 4 sin (x) - 1 < 0 - 1

Simplify

- 4 sin (x) < - 1

- 4 sin (x) < - 1

Multiply both sides by

- 1

- 4 sin (x) < - 1

Multiply both sides by -1 (reverse the inequality)

(- 4 sin (x)) (- 1) > (- 1) (- 1)

Simplify

4 sin (x) > 1

4 sin (x) > 1

Divide both sides by

4

4 sin (x) > 1

Divide both sides by

4

\frac{4 sin ( x )}{4} > \frac{1}{4}

Simplify

sin (x) > \frac{1}{4}

sin (x) > \frac{1}{4}

1 - 4 sin (x) > 0 : sin (x) < \frac{1}{4}

1 - 4 sin (x) > 0

Move

1

to the right side

1 - 4 sin (x) > 0

Subtract

1

from both sides

1 - 4 sin (x) - 1 > 0 - 1

Simplify

- 4 sin (x) > - 1

- 4 sin (x) > - 1

Multiply both sides by

- 1

- 4 sin (x) > - 1

Multiply both sides by -1 (reverse the inequality)

(- 4 sin (x)) (- 1) < (- 1) (- 1)

Simplify

4 sin (x) < 1

4 sin (x) < 1

Divide both sides by

4

4 sin (x) < 1

Divide both sides by

4

\frac{4 sin ( x )}{4} < \frac{1}{4}

Simplify

sin (x) < \frac{1}{4}

sin (x) < \frac{1}{4}

Find the signs of

sin (x)

sin (x) = 0

sin (x) < 0

sin (x) > 0

Find singularity points

Find the zeros of the denominator

sin (x) : sin (x) = 0

Summarize in a table:

1 - 4 sin (x) sin (x) \frac{1 - 4 s i n ( x )}{s i n ( x )} sin (x) < 0 + - - sin (x) = 0 + 0 Undefined 0 < sin (x) < \frac{1}{4} + + + sin (x) = \frac{1}{4} 0 + 0 sin (x) > \frac{1}{4} - + -

Identify the intervals that satisfy the required condition:

< 0

sin (x) < 0 or sin (x) > \frac{1}{4}

sin (x) < 0 or sin (x) > \frac{1}{4}

sin (x) < 0 : - π + 2 πn < x < 2 πn

sin (x) < 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 < x < arcsin (0) + 2 πn

Simplify

- π - arcsin (0) : - π

- π - arcsin (0)

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

- π - 0 = - π

= - π

Simplify

arcsin (0) : 0

arcsin (0)

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

- π + 2 πn < x < 0 + 2 πn

Simplify

- π + 2 πn < x < 2 πn

sin (x) > \frac{1}{4} : arcsin (\frac{1}{4}) + 2 πn < x < π - arcsin (\frac{1}{4}) + 2 πn

sin (x) > \frac{1}{4}

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (\frac{1}{4}) + 2 πn < x < π - arcsin (\frac{1}{4}) + 2 πn

Combine the intervals

- π + 2 πn < x < 2 πn or arcsin (\frac{1}{4}) + 2 πn < x < π - arcsin (\frac{1}{4}) + 2 πn

Merge Overlapping Intervals

arcsin (\frac{1}{4}) + 2 πn < x < π - arcsin (\frac{1}{4}) + 2 πn or - π + 2 πn < x < 2 πn

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