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1/(sin^2(x))>= 1

Solution

\frac{1}{sin ^{2} ( x )} \geq 1

Solution

2 πn < x < π + 2 πn or - π + 2 πn < x < 2 πn

Interval Notation

(2 πn, π + 2 πn) \cup (- π + 2 πn, 2 πn)

Decimal Notation

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

Solution steps

\frac{1}{sin ^{2} ( x )} \geq 1

Rewrite in standard form

\frac{1}{sin ^{2} ( x )} \geq 1

Subtract

1

from both sides

\frac{1}{sin ^{2} ( x )} - 1 \geq 1 - 1

Simplify

\frac{1}{sin ^{2} ( x )} - 1 \geq 0

Simplify

\frac{1}{sin ^{2} ( x )} - 1 : \frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )}

\frac{1}{sin ^{2} ( x )} - 1

Convert element to fraction:

1 = \frac{1 sin ^{2} ( x )}{sin ^{2} ( x )}

= \frac{1}{sin ^{2} ( x )} - \frac{1 \cdot sin ^{2} ( x )}{sin ^{2} ( 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 ^{2} ( x )}{sin ^{2} ( x )}

Multiply:

1 \cdot sin^{2} (x) = sin^{2} (x)

= \frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )}

\frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )} \geq 0

\frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )} \geq 0

Factor

\frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )} : \frac{- ( sin ( x ) + 1 ) ( sin ( x ) - 1 )}{sin ^{2} ( x )}

\frac{1 - sin ^{2} ( x )}{sin ^{2} ( x )}

Factor

- sin^{2} (x) + 1 : - (sin (x) + 1) (sin (x) - 1)

- sin^{2} (x) + 1

Factor out common term

- 1

= - (sin^{2} (x) - 1)

Factor

sin^{2} (x) - 1 : (sin (x) + 1) (sin (x) - 1)

sin^{2} (x) - 1

Rewrite

1

1^{2}

= sin^{2} (x) - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

sin^{2} (x) - 1^{2} = (sin (x) + 1) (sin (x) - 1)

= (sin (x) + 1) (sin (x) - 1)

= - (sin (x) + 1) (sin (x) - 1)

= \frac{- ( sin ( x ) + 1 ) ( sin ( x ) - 1 )}{sin ^{2} ( x )}

\frac{- ( sin ( x ) + 1 ) ( sin ( x ) - 1 )}{sin ^{2} ( x )} \geq 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( sin ( x ) + 1 ) ( sin ( x ) - 1 ) ) ( - 1 )}{sin ^{2} ( x )} \leq 0 \cdot (- 1)

Simplify

\frac{( sin ( x ) + 1 ) ( sin ( x ) - 1 )}{sin ^{2} ( x )} \leq 0

Identify the intervals

Find the signs of the factors of

\frac{( sin ( x ) + 1 ) ( sin ( x ) - 1 )}{sin ^{2} ( x )}

Find the signs of

sin (x) + 1

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

sin (x) + 1 = 0

Move

1

to the right side

sin (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

sin (x) = - 1

sin (x) = - 1

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

sin (x) + 1 < 0

Move

1

to the right side

sin (x) + 1 < 0

Subtract

1

from both sides

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

Simplify

sin (x) < - 1

sin (x) < - 1

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

sin (x) + 1 > 0

Move

1

to the right side

sin (x) + 1 > 0

Subtract

1

from both sides

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

Simplify

sin (x) > - 1

sin (x) > - 1

Find the signs of

sin (x) - 1

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

sin (x) - 1 = 0

Move

1

to the right side

sin (x) - 1 = 0

Add

1

to both sides

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

Simplify

sin (x) = 1

sin (x) = 1

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

sin (x) - 1 < 0

Move

1

to the right side

sin (x) - 1 < 0

Add

1

to both sides

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

Simplify

sin (x) < 1

sin (x) < 1

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

sin (x) - 1 > 0

Move

1

to the right side

sin (x) - 1 > 0

Add

1

to both sides

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

Simplify

sin (x) > 1

sin (x) > 1

Find the signs of

sin^{2} (x)

sin^{2} (x) = 0 : sin (x) = 0

sin^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

sin (x) = 0

sin^{2} (x) > 0 : sin (x) < 0 or sin (x) > 0

sin^{2} (x) > 0

For

u^{n} > 0

, if

n

is even then

u < 0

u > 0

sin (x) < 0 or sin (x) > 0

Find singularity points

Find the zeros of the denominator

sin^{2} (x) :

No Solution

sin^{2} (x) = 0

The sides are not equal

No Solution

Summarize in a table:

sin (x) + 1 sin (x) - 1 sin^{2} (x) \frac{( s i n ( x ) + 1 ) ( s i n ( x ) - 1 )}{s i n ^{2} ( x )} sin (x) < - 1 - - + + sin (x) = - 1 0 - + 0 - 1 < 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:

\leq 0

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

Merge Overlapping Intervals

- 1 \leq sin (x) < 0 or 0 < sin (x) < 1 or sin (x) = 1

The union of two intervals is the set of numbers which are in either interval

sin (x) = - 1

- 1 < sin (x) < 0

- 1 \leq sin (x) < 0

The union of two intervals is the set of numbers which are in either interval

- 1 \leq sin (x) < 0

0 < sin (x) < 1

- 1 \leq sin (x) < 0 or 0 < sin (x) < 1

The union of two intervals is the set of numbers which are in either interval

- 1 \leq sin (x) < 0 or 0 < sin (x) < 1

sin (x) = 1

- 1 \leq sin (x) < 0 or 0 < sin (x) \leq 1

- 1 \leq sin (x) < 0 or 0 < sin (x) \leq 1

- 1 \leq sin (x) < 0 or 0 < sin (x) \leq 1

- 1 \leq sin (x) < 0 : - π + 2 πn < x < 2 πn

- 1 \leq sin (x) < 0

a \leq u < b

then

a \leq u and u < b

- 1 \leq sin (x) and sin (x) < 0

- 1 \leq sin (x) :

True for all

x \in R

- 1 \leq sin (x)

Switch sides

sin (x) \geq - 1

Range of

sin (x) : - 1 \leq sin (x) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (x) \leq 1

- 1 \leq sin (x) \leq 1

sin (x) \geq - 1 and - 1 \leq sin (x) \leq 1 : - 1 \leq sin (x) \leq 1

Let

y = sin (x)

Combine the intervals

y \geq - 1 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \geq - 1 and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y \geq - 1

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

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

Combine the intervals

True for all x \in R and - π + 2 πn < x < 2 πn

Merge Overlapping Intervals

- π + 2 πn < x < 2 πn

0 < sin (x) \leq 1 : 2 πn < x < π + 2 πn

0 < sin (x) \leq 1

a < u \leq b

then

a < u and u \leq b

0 < sin (x) and sin (x) \leq 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) \leq 1 :

True for all

x \in R

sin (x) \leq 1

Range of

sin (x) : - 1 \leq sin (x) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (x) \leq 1

- 1 \leq sin (x) \leq 1

sin (x) \leq 1 and - 1 \leq sin (x) \leq 1 : - 1 \leq sin (x) \leq 1

Let

y = sin (x)

Combine the intervals

y \leq 1 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \leq 1 and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y \leq 1

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

Combine the intervals

2 πn < x < π + 2 πn and True for all x \in R

Merge Overlapping Intervals

2 πn < x < π + 2 πn

Combine the intervals

- π + 2 πn < x < 2 πn or 2 πn < x < π + 2 πn

Merge Overlapping Intervals

2 πn < x < π + 2 πn or - π + 2 πn < x < 2 πn

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