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Popular >

csc(x)>1

Solution

csc (x) > 1

Solution

2 πn < x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < π + 2 πn

Interval Notation

(2 πn, \frac{π}{2} + 2 πn) \cup (\frac{π}{2} + 2 πn, π + 2 πn)

Decimal Notation

2 πn < x < 1.57079 \dots + 2 πn or 1.57079 \dots + 2 πn < x < 3.14159 \dots + 2 πn

Solution steps

csc (x) > 1

Express with sin, cos

csc (x) > 1

Use the basic trigonometric identity:

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

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

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

Rewrite in standard form

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

Subtract

1

from both sides

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

Simplify

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

Simplify

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

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

Convert element to fraction:

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

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

Multiply:

1 \cdot sin (x) = sin (x)

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

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

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

Identify the intervals

Find the signs of the factors of

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

Find the signs of

1 - sin (x)

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

1 - sin (x) = 0

Move

1

to the right side

1 - sin (x) = 0

Subtract

1

from both sides

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

Simplify

- sin (x) = - 1

- sin (x) = - 1

Divide both sides by

- 1

- sin (x) = - 1

Divide both sides by

- 1

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

Simplify

sin (x) = 1

sin (x) = 1

1 - sin (x) < 0 : sin (x) > 1

1 - sin (x) < 0

Move

1

to the right side

1 - sin (x) < 0

Subtract

1

from both sides

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

Simplify

- sin (x) < - 1

- sin (x) < - 1

Multiply both sides by

- 1

- sin (x) < - 1

Multiply both sides by -1 (reverse the inequality)

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

Simplify

sin (x) > 1

sin (x) > 1

1 - sin (x) > 0 : sin (x) < 1

1 - sin (x) > 0

Move

1

to the right side

1 - sin (x) > 0

Subtract

1

from both sides

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

Simplify

- sin (x) > - 1

- sin (x) > - 1

Multiply both sides by

- 1

- sin (x) > - 1

Multiply both sides by -1 (reverse the inequality)

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

Simplify

sin (x) < 1

sin (x) < 1

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

Identify the intervals that satisfy the required condition:

> 0

0 < sin (x) < 1

0 < sin (x) < 1

a < u < b

then

a < u and u < b

0 < sin (x) and sin (x) < 1

0 < sin (x) : 2 πn < x < π + 2 πn

0 < sin (x)

Switch sides

sin (x) > 0

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

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

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

0 + 2 πn < x < π + 2 πn

Simplify

2 πn < x < π + 2 πn

sin (x) < 1 : - \frac{3 π}{2} + 2 πn < x < \frac{π}{2} + 2 πn

sin (x) < 1

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (1) + 2 πn < x < arcsin (1) + 2 πn

Simplify

- π - arcsin (1) : - \frac{3 π}{2}

- π - arcsin (1)

Use the following trivial identity:

arcsin (1) = \frac{π}{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}

= - π - \frac{π}{2}

Simplify

- 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 = - 3 π

= \frac{- 3 π}{2}

Apply the fraction rule:

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

= - \frac{3 π}{2}

= - \frac{3 π}{2}

Simplify

arcsin (1) : \frac{π}{2}

arcsin (1)

Use the following trivial identity:

arcsin (1) = \frac{π}{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}{2}

- \frac{3 π}{2} + 2 πn < x < \frac{π}{2} + 2 πn

Combine the intervals

2 πn < x < π + 2 πn and - \frac{3 π}{2} + 2 πn < x < \frac{π}{2} + 2 πn

Merge Overlapping Intervals

2 πn < x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < π + 2 πn

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