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Popular Trigonometry >

2sin^4(x)-3sin^2(x)+1>0

Solution

2 sin^{4} (x) - 3 sin^{2} (x) + 1 > 0

Solution

2 πn \leq x < \frac{π}{4} + 2 πn or \frac{3 π}{4} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{7 π}{4} + 2 πn < x < 2 π + 2 πn

Interval Notation

[2 πn, \frac{π}{4} + 2 πn) \cup (\frac{3 π}{4} + 2 πn, \frac{5 π}{4} + 2 πn) \cup (\frac{7 π}{4} + 2 πn, 2 π + 2 πn)

Decimal

2 πn \leq x < 0.78539 \dots + 2 πn or 2.35619 \dots + 2 πn < x < 3.92699 \dots + 2 πn or 5.49778 \dots + 2 πn < x < 6.28318 \dots + 2 πn

Solution steps

2 sin^{4} (x) - 3 sin^{2} (x) + 1 > 0

Let:

v = sin (x)

2 v^{4} - 3 v^{2} + 1 > 0

2 v^{4} - 3 v^{2} + 1 > 0 : v < - 1 or - \frac{2}{2} < v < \frac{2}{2} or v > 1

2 v^{4} - 3 v^{2} + 1 > 0

Factor

2 v^{4} - 3 v^{2} + 1 : (2 v + 1) (2 v - 1) (v + 1) (v - 1)

2 v^{4} - 3 v^{2} + 1

Let

u = v^{2}

= 2 u^{2} - 3 u + 1

Factor

2 u^{2} - 3 u + 1 : (2 u - 1) (u - 1)

2 u^{2} - 3 u + 1

Break the expression into groups

2 u^{2} - 3 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 = - 3

Check

u = 1, v = 2 : u * v = 2, u + v = 3 \Rightarrow

FalseCheck

u = - 1, v = - 2 : u * v = 2, u + v = - 3 \Rightarrow

True

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)

Substitute back

u = v^{2}

= (v^{2} - 1) (2 v^{2} - 1)

Factor

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

2 v^{2} - 1

Rewrite

2 v^{2} - 1

(2 v)^{2} - 1^{2}

2 v^{2} - 1

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} v^{2} - 1

Rewrite

1

1^{2}

= (2)^{2} v^{2} - 1^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(2)^{2} v^{2} = (2 v)^{2}

= (2 v)^{2} - 1^{2}

= (2 v)^{2} - 1^{2}

Apply Difference of Two Squares Formula:

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

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

= (2 v + 1) (2 v - 1)

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

Factor

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

v^{2} - 1

Rewrite

1

1^{2}

= v^{2} - 1^{2}

Apply Difference of Two Squares Formula:

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

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

= (v + 1) (v - 1)

= (2 v + 1) (2 v - 1) (v + 1) (v - 1)

(2 v + 1) (2 v - 1) (v + 1) (v - 1) > 0

Identify the intervals

Find the signs of the factors of

(2 v + 1) (2 v - 1) (v + 1) (v - 1)

Find the signs of

2 v + 1

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

2 v + 1 = 0

Move

1

to the right side

2 v + 1 = 0

Subtract

1

from both sides

2 v + 1 - 1 = 0 - 1

Simplify

2 v = - 1

2 v = - 1

Divide both sides by

2

2 v = - 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 v}{2} : v

\frac{2 v}{2}

Cancel the common factor:

2

= v

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

v = - \frac{2}{2}

v = - \frac{2}{2}

v = - \frac{2}{2}

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

2 v + 1 < 0

Move

1

to the right side

2 v + 1 < 0

Subtract

1

from both sides

2 v + 1 - 1 < 0 - 1

Simplify

2 v < - 1

2 v < - 1

Divide both sides by

2

2 v < - 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 v}{2} : v

\frac{2 v}{2}

Cancel the common factor:

2

= v

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

v < - \frac{2}{2}

v < - \frac{2}{2}

v < - \frac{2}{2}

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

2 v + 1 > 0

Move

1

to the right side

2 v + 1 > 0

Subtract

1

from both sides

2 v + 1 - 1 > 0 - 1

Simplify

2 v > - 1

2 v > - 1

Divide both sides by

2

2 v > - 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 v}{2} : v

\frac{2 v}{2}

Cancel the common factor:

2

= v

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

v > - \frac{2}{2}

v > - \frac{2}{2}

v > - \frac{2}{2}

Find the signs of

2 v - 1

2 v - 1 = 0 : v = \frac{2}{2}

2 v - 1 = 0

Move

1

to the right side

2 v - 1 = 0

Add

1

to both sides

2 v - 1 + 1 = 0 + 1

Simplify

2 v = 1

2 v = 1

Divide both sides by

2

2 v = 1

Divide both sides by

2

\frac{2 v}{2} = \frac{1}{2}

Simplify

\frac{2 v}{2} = \frac{1}{2}

Simplify

\frac{2 v}{2} : v

\frac{2 v}{2}

Cancel the common factor:

2

= v

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

v = \frac{2}{2}

v = \frac{2}{2}

v = \frac{2}{2}

2 v - 1 < 0 : v < \frac{2}{2}

2 v - 1 < 0

Move

1

to the right side

2 v - 1 < 0

Add

1

to both sides

2 v - 1 + 1 < 0 + 1

Simplify

2 v < 1

2 v < 1

Divide both sides by

2

2 v < 1

Divide both sides by

2

\frac{2 v}{2} < \frac{1}{2}

Simplify

\frac{2 v}{2} < \frac{1}{2}

Simplify

\frac{2 v}{2} : v

\frac{2 v}{2}

Cancel the common factor:

2

= v

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

v < \frac{2}{2}

v < \frac{2}{2}

v < \frac{2}{2}

2 v - 1 > 0 : v > \frac{2}{2}

2 v - 1 > 0

Move

1

to the right side

2 v - 1 > 0

Add

1

to both sides

2 v - 1 + 1 > 0 + 1

Simplify

2 v > 1

2 v > 1

Divide both sides by

2

2 v > 1

Divide both sides by

2

\frac{2 v}{2} > \frac{1}{2}

Simplify

\frac{2 v}{2} > \frac{1}{2}

Simplify

\frac{2 v}{2} : v

\frac{2 v}{2}

Cancel the common factor:

2

= v

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

v > \frac{2}{2}

v > \frac{2}{2}

v > \frac{2}{2}

Find the signs of

v + 1

v + 1 = 0 : v = - 1

v + 1 = 0

Move

1

to the right side

v + 1 = 0

Subtract

1

from both sides

v + 1 - 1 = 0 - 1

Simplify

v = - 1

v = - 1

v + 1 < 0 : v < - 1

v + 1 < 0

Move

1

to the right side

v + 1 < 0

Subtract

1

from both sides

v + 1 - 1 < 0 - 1

Simplify

v < - 1

v < - 1

v + 1 > 0 : v > - 1

v + 1 > 0

Move

1

to the right side

v + 1 > 0

Subtract

1

from both sides

v + 1 - 1 > 0 - 1

Simplify

v > - 1

v > - 1

Find the signs of

v - 1

v - 1 = 0 : v = 1

v - 1 = 0

Move

1

to the right side

v - 1 = 0

Add

1

to both sides

v - 1 + 1 = 0 + 1

Simplify

v = 1

v = 1

v - 1 < 0 : v < 1

v - 1 < 0

Move

1

to the right side

v - 1 < 0

Add

1

to both sides

v - 1 + 1 < 0 + 1

Simplify

v < 1

v < 1

v - 1 > 0 : v > 1

v - 1 > 0

Move

1

to the right side

v - 1 > 0

Add

1

to both sides

v - 1 + 1 > 0 + 1

Simplify

v > 1

v > 1

Summarize in a table:

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

Identify the intervals that satisfy the required condition:

> 0

v < - 1 or - \frac{2}{2} < v < \frac{2}{2} or v > 1

v < - 1 or - \frac{2}{2} < v < \frac{2}{2} or v > 1

v < - 1 or - \frac{2}{2} < v < \frac{2}{2} or v > 1

Substitute back

v = sin (x)

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

sin (x) < - 1 :

False for all

x \in R

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

False

Let

y = sin (x)

Combine the intervals

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

Merge Overlapping Intervals

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

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

y < - 1

and

- 1 \leq y \leq 1

False for all y \in R

False for all y \in R

No Solution for x \in R

False for all x \in R

- \frac{2}{2} < sin (x) < \frac{2}{2} : 2 πn \leq x < \frac{π}{4} + 2 πn or \frac{3 π}{4} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{7 π}{4} + 2 πn < x < 2 π + 2 πn

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

a < u < b

then

a < u and u < b

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

- \frac{2}{2} < sin (x) : - \frac{π}{4} + 2 πn < x < \frac{5 π}{4} + 2 πn

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

Switch sides

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

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

Simplify

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

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

Use the following property:

arcsin (- x) = - arcsin (x)

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

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

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

arcsin (\frac{2}{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}{4}

= - \frac{π}{4}

Simplify

π - arcsin (- \frac{2}{2}) : \frac{5 π}{4}

π - arcsin (- \frac{2}{2})

arcsin (- \frac{2}{2}) = - \frac{π}{4}

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

Use the following property:

arcsin (- x) = - arcsin (x)

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

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

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

arcsin (\frac{2}{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}{4}

= - \frac{π}{4}

= π - (- \frac{π}{4})

Simplify

π - (- \frac{π}{4})

Apply rule

- (- a) = a

= π + \frac{π}{4}

Convert element to fraction:

π = \frac{π 4}{4}

= \frac{π 4}{4} + \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 + π}{4}

Add similar elements:

4 π + π = 5 π

= \frac{5 π}{4}

= \frac{5 π}{4}

- \frac{π}{4} + 2 πn < x < \frac{5 π}{4} + 2 πn

sin (x) < \frac{2}{2} : - \frac{5 π}{4} + 2 πn < x < \frac{π}{4} + 2 πn

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

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

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

Simplify

- π - arcsin (\frac{2}{2}) : - \frac{5 π}{4}

- π - arcsin (\frac{2}{2})

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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}{4}

Simplify

- π - \frac{π}{4}

Convert element to fraction:

π = \frac{π 4}{4}

= - \frac{π 4}{4} - \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 4 - π}{4}

Add similar elements:

- 4 π - π = - 5 π

= \frac{- 5 π}{4}

Apply the fraction rule:

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

= - \frac{5 π}{4}

= - \frac{5 π}{4}

Simplify

arcsin (\frac{2}{2}) : \frac{π}{4}

arcsin (\frac{2}{2})

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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}{4}

- \frac{5 π}{4} + 2 πn < x < \frac{π}{4} + 2 πn

Combine the intervals

- \frac{π}{4} + 2 πn < x < \frac{5 π}{4} + 2 πn and - \frac{5 π}{4} + 2 πn < x < \frac{π}{4} + 2 πn

Merge Overlapping Intervals

2 πn \leq x < \frac{π}{4} + 2 πn or \frac{3 π}{4} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{7 π}{4} + 2 πn < x < 2 π + 2 πn

sin (x) > 1 :

False for all

x \in R

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

False

Let

y = sin (x)

Combine the intervals

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

Merge Overlapping Intervals

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

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

y > 1

and

- 1 \leq y \leq 1

False for all y \in R

False for all y \in R

No Solution for x \in R

False for all x \in R

Combine the intervals

False for all x \in R or (2 πn \leq x < \frac{π}{4} + 2 πn or \frac{3 π}{4} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{7 π}{4} + 2 πn < x < 2 π + 2 πn) or False for all x \in R

Merge Overlapping Intervals

2 πn \leq x < \frac{π}{4} + 2 πn or \frac{3 π}{4} + 2 πn < x < \frac{5 π}{4} + 2 πn or \frac{7 π}{4} + 2 πn < x < 2 π + 2 πn

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