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Popular Trigonometry >

(sin(x)+4)(sqrt(3)-2sin(x))>0

Solution

(sin (x) + 4) (3 - 2 sin (x)) > 0

Solution

- \frac{4 π}{3} + 2 πn < x < \frac{π}{3} + 2 πn

Interval Notation

(- \frac{4 π}{3} + 2 πn, \frac{π}{3} + 2 πn)

Decimal

- 4.18879 \dots + 2 πn < x < 1.04719 \dots + 2 πn

Solution steps

(sin (x) + 4) (3 - 2 sin (x)) > 0

Let:

u = sin (x)

(u + 4) (3 - 2 u) > 0

(u + 4) (3 - 2 u) > 0 : - 4 < u < \frac{3}{2}

(u + 4) (3 - 2 u) > 0

Identify the intervals

Find the signs of the factors of

(u + 4) (3 - 2 u)

Find the signs of

u + 4

u + 4 = 0 : u = - 4

u + 4 = 0

Move

4

to the right side

u + 4 = 0

Subtract

4

from both sides

u + 4 - 4 = 0 - 4

Simplify

u = - 4

u = - 4

u + 4 < 0 : u < - 4

u + 4 < 0

Move

4

to the right side

u + 4 < 0

Subtract

4

from both sides

u + 4 - 4 < 0 - 4

Simplify

u < - 4

u < - 4

u + 4 > 0 : u > - 4

u + 4 > 0

Move

4

to the right side

u + 4 > 0

Subtract

4

from both sides

u + 4 - 4 > 0 - 4

Simplify

u > - 4

u > - 4

Find the signs of

3 - 2 u

3 - 2 u = 0 : u = \frac{3}{2}

3 - 2 u = 0

Move

3

to the right side

3 - 2 u = 0

Subtract

3

from both sides

3 - 2 u - 3 = 0 - 3

Simplify

- 2 u = - 3

- 2 u = - 3

Divide both sides by

- 2

- 2 u = - 3

Divide both sides by

- 2

\frac{- 2 u}{- 2} = \frac{- 3}{- 2}

Simplify

u = \frac{3}{2}

u = \frac{3}{2}

3 - 2 u < 0 : u > \frac{3}{2}

3 - 2 u < 0

Move

3

to the right side

3 - 2 u < 0

Subtract

3

from both sides

3 - 2 u - 3 < 0 - 3

Simplify

- 2 u < - 3

- 2 u < - 3

Multiply both sides by

- 1

- 2 u < - 3

Multiply both sides by -1 (reverse the inequality)

(- 2 u) (- 1) > (- 3) (- 1)

Simplify

2 u > 3

2 u > 3

Divide both sides by

2

2 u > 3

Divide both sides by

2

\frac{2 u}{2} > \frac{3}{2}

Simplify

u > \frac{3}{2}

u > \frac{3}{2}

3 - 2 u > 0 : u < \frac{3}{2}

3 - 2 u > 0

Move

3

to the right side

3 - 2 u > 0

Subtract

3

from both sides

3 - 2 u - 3 > 0 - 3

Simplify

- 2 u > - 3

- 2 u > - 3

Multiply both sides by

- 1

- 2 u > - 3

Multiply both sides by -1 (reverse the inequality)

(- 2 u) (- 1) < (- 3) (- 1)

Simplify

2 u < 3

2 u < 3

Divide both sides by

2

2 u < 3

Divide both sides by

2

\frac{2 u}{2} < \frac{3}{2}

Simplify

u < \frac{3}{2}

u < \frac{3}{2}

Summarize in a table:

u + 4 3 - 2 u (u + 4) (3 - 2 u) u < - 4 - + - u = - 4 0 + 0 - 4 < u < \frac{3}{2} + + + u = \frac{3}{2} + 00 u > \frac{3}{2} + - -

Identify the intervals that satisfy the required condition:

> 0

- 4 < u < \frac{3}{2}

- 4 < u < \frac{3}{2}

- 4 < u < \frac{3}{2}

Substitute back

u = sin (x)

- 4 < sin (x) < \frac{3}{2}

a < u < b

then

a < u and u < b

- 4 < sin (x) and sin (x) < \frac{3}{2}

- 4 < sin (x) :

True for all

x \in R

- 4 < sin (x)

Switch sides

sin (x) > - 4

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) > - 4 and - 1 \leq sin (x) \leq 1 : - 1 \leq sin (x) \leq 1

Let

y = sin (x)

Combine the intervals

y > - 4 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y > - 4 and - 1 \leq y \leq 1

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

y > - 4

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) < \frac{3}{2} : - \frac{4 π}{3} + 2 πn < x < \frac{π}{3} + 2 πn

sin (x) < \frac{3}{2}

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

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

Simplify

- π - arcsin (\frac{3}{2}) : - \frac{4 π}{3}

- π - arcsin (\frac{3}{2})

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

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}{3}

Simplify

- π - \frac{π}{3}

Convert element to fraction:

π = \frac{π 3}{3}

= - \frac{π 3}{3} - \frac{π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 - π}{3}

Add similar elements:

- 3 π - π = - 4 π

= \frac{- 4 π}{3}

Apply the fraction rule:

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

= - \frac{4 π}{3}

= - \frac{4 π}{3}

Simplify

arcsin (\frac{3}{2}) : \frac{π}{3}

arcsin (\frac{3}{2})

Use the following trivial identity:

arcsin (\frac{3}{2}) = \frac{π}{3}

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}{3}

- \frac{4 π}{3} + 2 πn < x < \frac{π}{3} + 2 πn

Combine the intervals

True for all x \in R and - \frac{4 π}{3} + 2 πn < x < \frac{π}{3} + 2 πn

Merge Overlapping Intervals

- \frac{4 π}{3} + 2 πn < x < \frac{π}{3} + 2 πn

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