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Popular Trigonometry >

2>(24)/(sin(θ))

Solution

2 > \frac{24}{sin ( θ )}

Solution

- π + 2 πn < θ < 2 πn

Interval Notation

(- π + 2 πn, 2 πn)

Decimal

- 3.14159 \dots + 2 πn < θ < 2 πn

Solution steps

2 > \frac{24}{sin ( θ )}

Switch sides

\frac{24}{sin ( θ )} < 2

Rewrite in standard form

\frac{24}{sin ( θ )} < 2

Subtract

2

from both sides

\frac{24}{sin ( θ )} - 2 < 2 - 2

Simplify

\frac{24}{sin ( θ )} - 2 < 0

Simplify

\frac{24}{sin ( θ )} - 2 : \frac{24 - 2 sin ( θ )}{sin ( θ )}

\frac{24}{sin ( θ )} - 2

Convert element to fraction:

2 = \frac{2 sin ( θ )}{sin ( θ )}

= \frac{24}{sin ( θ )} - \frac{2 sin ( θ )}{sin ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{24 - 2 sin ( θ )}{sin ( θ )}

\frac{24 - 2 sin ( θ )}{sin ( θ )} < 0

\frac{24 - 2 sin ( θ )}{sin ( θ )} < 0

Factor

\frac{24 - 2 sin ( θ )}{sin ( θ )} : \frac{- 2 ( sin ( θ ) - 12 )}{sin ( θ )}

\frac{24 - 2 sin ( θ )}{sin ( θ )}

Factor

- 2 sin (θ) + 24 : - 2 (sin (θ) - 12)

- 2 sin (θ) + 24

Factor out common term

- 2 : - 2 (sin (θ) - 12)

- 2 sin (θ) + 24

Rewrite

24

2 \cdot 12

= - 2 sin (θ) + 2 \cdot 12

Factor out common term

- 2

= - 2 (sin (θ) - 12)

= - 2 (sin (θ) - 12)

= \frac{- 2 ( sin ( θ ) - 12 )}{sin ( θ )}

\frac{- 2 ( sin ( θ ) - 12 )}{sin ( θ )} < 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - 2 ( sin ( θ ) - 12 ) ) ( - 1 )}{sin ( θ )} > 0 \cdot (- 1)

Simplify

\frac{2 ( sin ( θ ) - 12 )}{sin ( θ )} > 0

Divide both sides by

2

\frac{\frac{2 ( s i n ( θ ) - 12 )}{s i n ( θ )}}{2} > \frac{0}{2}

Simplify

\frac{sin ( θ ) - 12}{sin ( θ )} > 0

Identify the intervals

Find the signs of the factors of

\frac{sin ( θ ) - 12}{sin ( θ )}

Find the signs of

sin (θ) - 12

sin (θ) - 12 = 0 : sin (θ) = 12

sin (θ) - 12 = 0

Move

12

to the right side

sin (θ) - 12 = 0

Add

12

to both sides

sin (θ) - 12 + 12 = 0 + 12

Simplify

sin (θ) = 12

sin (θ) = 12

sin (θ) - 12 < 0 : sin (θ) < 12

sin (θ) - 12 < 0

Move

12

to the right side

sin (θ) - 12 < 0

Add

12

to both sides

sin (θ) - 12 + 12 < 0 + 12

Simplify

sin (θ) < 12

sin (θ) < 12

sin (θ) - 12 > 0 : sin (θ) > 12

sin (θ) - 12 > 0

Move

12

to the right side

sin (θ) - 12 > 0

Add

12

to both sides

sin (θ) - 12 + 12 > 0 + 12

Simplify

sin (θ) > 12

sin (θ) > 12

Find the signs of

sin (θ)

sin (θ) = 0

sin (θ) < 0

sin (θ) > 0

Find singularity points

Find the zeros of the denominator

sin (θ) : sin (θ) = 0

Summarize in a table:

sin (θ) - 12 sin (θ) \frac{s i n ( θ ) - 12}{s i n ( θ )} sin (θ) < 0 - - + sin (θ) = 0 - 0 Undefined 0 < sin (θ) < 12 - + - sin (θ) = 12 0 + 0 sin (θ) > 12 + + +

Identify the intervals that satisfy the required condition:

> 0

sin (θ) < 0 or sin (θ) > 12

sin (θ) < 0 or sin (θ) > 12

sin (θ) < 0 : - π + 2 πn < θ < 2 πn

sin (θ) < 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 < θ < 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 < θ < 0 + 2 πn

Simplify

- π + 2 πn < θ < 2 πn

sin (θ) > 12 :

False for all

θ \in R

sin (θ) > 12

Range of

sin (θ) : - 1 \leq sin (θ) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (θ) \leq 1

- 1 \leq sin (θ) \leq 1

sin (θ) > 12 and - 1 \leq sin (θ) \leq 1 :

False

Let

y = sin (θ)

Combine the intervals

y > 12 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y > 12 and - 1 \leq y \leq 1

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

y > 12

and

- 1 \leq y \leq 1

False for all y \in R

False for all y \in R

No Solution for θ \in R

False for all θ \in R

Combine the intervals

- π + 2 πn < θ < 2 πn or False for all θ \in R

Merge Overlapping Intervals

- π + 2 πn < θ < 2 πn

Popular Examples

5sin(1/2 (x+pi/4))-1>= 7