English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

2sin^2(x)-sin(x)-1<0

Solution

2 sin^{2} (x) - sin (x) - 1 < 0

Solution

2 πn \leq x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < \frac{7 π}{6} + 2 πn or \frac{11 π}{6} + 2 πn < x < 2 π + 2 πn

Interval Notation

[2 πn, \frac{π}{2} + 2 πn) \cup (\frac{π}{2} + 2 πn, \frac{7 π}{6} + 2 πn) \cup (\frac{11 π}{6} + 2 πn, 2 π + 2 πn)

Decimal Notation

2 πn \leq x < 1.57079 \dots + 2 πn or 1.57079 \dots + 2 πn < x < 3.66519 \dots + 2 πn or 5.75958 \dots + 2 πn < x < 6.28318 \dots + 2 πn

Solution steps

2 sin^{2} (x) - sin (x) - 1 < 0

Let:

u = sin (x)

2 u^{2} - u - 1 < 0

2 u^{2} - u - 1 < 0 : - \frac{1}{2} < u < 1

2 u^{2} - u - 1 < 0

Factor

2 u^{2} - u - 1 : (2 u + 1) (u - 1)

2 u^{2} - u - 1

Break the expression into groups

2 u^{2} - u - 1

Definition

Factors of

2 : 1, 2

2

Divisors (Factors)

Find the Prime factors of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Add 1

1

The factors of

2

1, 2

Negative factors of

2 : - 1, - 2

Multiply the factors by

- 1

to get the negative factors

- 1, - 2

For every two factors such that

u * v = - 2,

check if

u + v = - 1

Check

u = 1, v = - 2 : u * v = - 2, u + v = - 1 \Rightarrow

TrueCheck

u = 2, v = - 1 : u * v = - 2, u + v = 1 \Rightarrow

False

u = 1, v = - 2

Group into

(a x^{2} + ux) + (vx + c)

(2 u^{2} + u) + (- 2 u - 1)

= (2 u^{2} + u) + (- 2 u - 1)

Factor out

u

from

2 u^{2} + u : u (2 u + 1)

2 u^{2} + u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{2} = uu

= 2 uu + u

Factor out common term

u

= u (2 u + 1)

Factor out

- 1

from

- 2 u - 1 : - (2 u + 1)

- 2 u - 1

Factor out common term

- 1

= - (2 u + 1)

= u (2 u + 1) - (2 u + 1)

Factor out common term

2 u + 1

= (2 u + 1) (u - 1)

(2 u + 1) (u - 1) < 0

Identify the intervals

Find the signs of the factors of

(2 u + 1) (u - 1)

Find the signs of

2 u + 1

2 u + 1 = 0 : u = - \frac{1}{2}

2 u + 1 = 0

Move

1

to the right side

2 u + 1 = 0

Subtract

1

from both sides

2 u + 1 - 1 = 0 - 1

Simplify

2 u = - 1

2 u = - 1

Divide both sides by

2

2 u = - 1

Divide both sides by

2

\frac{2 u}{2} = \frac{- 1}{2}

Simplify

u = - \frac{1}{2}

u = - \frac{1}{2}

2 u + 1 < 0 : u < - \frac{1}{2}

2 u + 1 < 0

Move

1

to the right side

2 u + 1 < 0

Subtract

1

from both sides

2 u + 1 - 1 < 0 - 1

Simplify

2 u < - 1

2 u < - 1

Divide both sides by

2

2 u < - 1

Divide both sides by

2

\frac{2 u}{2} < \frac{- 1}{2}

Simplify

u < - \frac{1}{2}

u < - \frac{1}{2}

2 u + 1 > 0 : u > - \frac{1}{2}

2 u + 1 > 0

Move

1

to the right side

2 u + 1 > 0

Subtract

1

from both sides

2 u + 1 - 1 > 0 - 1

Simplify

2 u > - 1

2 u > - 1

Divide both sides by

2

2 u > - 1

Divide both sides by

2

\frac{2 u}{2} > \frac{- 1}{2}

Simplify

u > - \frac{1}{2}

u > - \frac{1}{2}

Find the signs of

u - 1

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to both sides

u - 1 + 1 = 0 + 1

Simplify

u = 1

u = 1

u - 1 < 0 : u < 1

u - 1 < 0

Move

1

to the right side

u - 1 < 0

Add

1

to both sides

u - 1 + 1 < 0 + 1

Simplify

u < 1

u < 1

u - 1 > 0 : u > 1

u - 1 > 0

Move

1

to the right side

u - 1 > 0

Add

1

to both sides

u - 1 + 1 > 0 + 1

Simplify

u > 1

u > 1

Summarize in a table:

2 u + 1 u - 1 (2 u + 1) (u - 1) u < - \frac{1}{2} - - + u = - \frac{1}{2} 0 - 0 - \frac{1}{2} < u < 1 + - - u = 1 + 00 u > 1 + + +

Identify the intervals that satisfy the required condition:

< 0

- \frac{1}{2} < u < 1

- \frac{1}{2} < u < 1

- \frac{1}{2} < u < 1

Substitute back

u = sin (x)

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

a < u < b

then

a < u and u < b

- \frac{1}{2} < sin (x) and sin (x) < 1

- \frac{1}{2} < sin (x) : - \frac{π}{6} + 2 πn < x < \frac{7 π}{6} + 2 πn

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

Switch sides

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

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

Simplify

arcsin (- \frac{1}{2}) : - \frac{π}{6}

arcsin (- \frac{1}{2})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{2}) = - arcsin (\frac{1}{2})

= - arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{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}{6}

= - \frac{π}{6}

Simplify

π - arcsin (- \frac{1}{2}) : \frac{7 π}{6}

π - arcsin (- \frac{1}{2})

arcsin (- \frac{1}{2}) = - \frac{π}{6}

arcsin (- \frac{1}{2})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{2}) = - arcsin (\frac{1}{2})

= - arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

arcsin (\frac{1}{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}{6}

= - \frac{π}{6}

= π - (- \frac{π}{6})

Simplify

π - (- \frac{π}{6})

Apply rule

- (- a) = a

= π + \frac{π}{6}

Convert element to fraction:

π = \frac{π 6}{6}

= \frac{π 6}{6} + \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 6 + π}{6}

Add similar elements:

6 π + π = 7 π

= \frac{7 π}{6}

= \frac{7 π}{6}

- \frac{π}{6} + 2 πn < x < \frac{7 π}{6} + 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

- \frac{π}{6} + 2 πn < x < \frac{7 π}{6} + 2 πn and - \frac{3 π}{2} + 2 πn < x < \frac{π}{2} + 2 πn

Merge Overlapping Intervals

2 πn \leq x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < \frac{7 π}{6} + 2 πn or \frac{11 π}{6} + 2 πn < x < 2 π + 2 πn

Popular Examples

cos^2(x/3)<= sin^2(x/3)-1/2

cos^{2} (\frac{x}{3}) \leq sin^{2} (\frac{x}{3}) - \frac{1}{2}

pi/2-arctan(e^x)>0.01

\frac{π}{2} - arctan (e^{x}) > 0.01

(cos(x)-1)>= 0

(cos (x) - 1) \geq 0

2sin(pix)>1

2 sin (π x) > 1

sin^2(x)cos(x)-cos(x)>= 0

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