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4sin^2(x)-2(1+sqrt(2))sin(x)+sqrt(2)<0

Solution

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

Solution

\frac{π}{6} + 2 πn < x < \frac{π}{4} + 2 πn or \frac{3 π}{4} + 2 πn < x < \frac{5 π}{6} + 2 πn

Interval Notation

(\frac{π}{6} + 2 πn, \frac{π}{4} + 2 πn) \cup (\frac{3 π}{4} + 2 πn, \frac{5 π}{6} + 2 πn)

Decimal Notation

0.52359 \dots + 2 πn < x < 0.78539 \dots + 2 πn or 2.35619 \dots + 2 πn < x < 2.61799 \dots + 2 πn

Solution steps

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

Let:

u = sin (x)

4 u^{2} - 2 (1 + 2) u + 2 < 0

4 u^{2} - 2 (1 + 2) u + 2 < 0 : \frac{1}{2} < u < \frac{2}{2}

4 u^{2} - 2 (1 + 2) u + 2 < 0

Complete the square

4 u^{2} - 2 (1 + 2) u + 2 : 4 u^{2} - 2 u - 22 u + 2

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

Write in the standard form

a x^{2} + b x + c

4 u^{2} - (2 + 22) u + 2

Write

4 u^{2} - (2 + 22) u + 2

in the form:

x^{2} + 2 a x + a^{2}

Factor out

4

4 (\frac{4 u ^{2}}{4} - \frac{( 2 + 2 2 ) u}{4} + \frac{2}{4})

Expand

\frac{4 u ^{2}}{4} - \frac{( 2 + 2 2 ) u}{4} + \frac{2}{4} : u^{2} - \frac{u}{2} - \frac{u}{2} + \frac{1}{2 2}

\frac{4 u ^{2}}{4} - \frac{( 2 + 2 2 ) u}{4} + \frac{2}{4}

Apply rule

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

= \frac{4 u ^{2} - ( 2 + 2 2 ) u + 2}{4}

Expand

4 u^{2} - (2 + 22) u + 2 : 4 u^{2} - 2 u - 22 u + 2

4 u^{2} - (2 + 22) u + 2

Expand

- u (2 + 22) : - 2 u - 22 u

- u (2 + 22)

Apply the distributive law:

a (b + c) = ab + a c

a = - u, b = 2, c = 22

= - u \cdot 2 + (- u) \cdot 22

Apply minus-plus rules

+ (- a) = - a

= - 2 u - 22 u

= 4 u^{2} - 2 u - 22 u + 2

= \frac{4 u ^{2} - 2 u - 2 2 u + 2}{4}

Apply the fraction rule:

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

\frac{4 u ^{2} - 2 u - 2 2 u + 2}{4} = \frac{4 u ^{2}}{4} - \frac{2 u}{4} - \frac{2 2 u}{4} + \frac{2}{4}

= \frac{4 u ^{2}}{4} - \frac{2 u}{4} - \frac{2 2 u}{4} + \frac{2}{4}

\frac{4 u ^{2}}{4} = u^{2}

\frac{4 u ^{2}}{4}

Divide the numbers:

\frac{4}{4} = 1

= u^{2}

\frac{2 u}{4} = \frac{u}{2}

\frac{2 u}{4}

Cancel the common factor:

2

= \frac{u}{2}

\frac{2 2 u}{4} = \frac{u}{2}

\frac{2 2 u}{4}

Cancel

\frac{2 2 u}{4} : \frac{u}{2}

\frac{2 2 u}{4}

Cancel the common factor:

2

= \frac{2 u}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

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

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{u}{2}

= \frac{u}{2}

\frac{2}{4} = \frac{1}{2 2}

\frac{2}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{2}{2 ^{2}}

Cancel

\frac{2}{2 ^{2}} : \frac{1}{2 ^{\frac{3}{2}}}

\frac{2}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{1}{2 ^{\frac{3}{2}}}

= \frac{1}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 22

= \frac{1}{2 2}

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

2 a = - \frac{1}{2} - \frac{1}{2} : a = - \frac{1 + 2}{4}

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 a}{2} : a

\frac{2 a}{2}

Divide the numbers:

\frac{2}{2} = 1

= a

Simplify

- \frac{\frac{1}{2}}{2} - \frac{\frac{1}{2}}{2} : - \frac{1 + 2}{4}

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

Apply rule

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

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

Join

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

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

Least Common Multiplier of

2, 2 : 22

2, 2

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

2

2

= 22

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

22

For

\frac{1}{2} :

multiply the denominator and numerator by

2

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

For

\frac{1}{2} :

multiply the denominator and numerator by

2

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

= - \frac{2}{2 2} - \frac{2}{2 2}

Since the denominators are equal, combine the fractions:

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

= \frac{- 2 - 2}{2 2}

Factor

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

- 2 - 2

2 = 22

= - 2 - 22

Factor out common term

2

= - 2 (1 + 2)

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

Cancel

- \frac{2 ( 1 + 2 )}{2 2} : - \frac{1 + 2}{2 2}

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

Apply radical rule:

2 = 2^{\frac{1}{2}}

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

= - \frac{1 + 2}{2 ^{\frac{1}{2}} 2}

Apply radical rule:

2^{\frac{1}{2}} = 2

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

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

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

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

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

Apply the fraction rule:

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

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

Apply the fraction rule:

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

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

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

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{1 + 2}{4}

a = - \frac{1 + 2}{4}

a = - \frac{1 + 2}{4}

a = - \frac{1 + 2}{4}

Add and subtract

(- \frac{1 + 2}{4})^{2}

4 (u^{2} - \frac{u}{2} - \frac{u}{2} + \frac{1}{2 2})

x^{2} + 2 a x + a^{2} = (x + a)^{2}

u^{2} - \frac{1}{2} - \frac{1}{2} u + (- \frac{1 + 2}{4})^{2} = (u - \frac{1 + 2}{4})^{2}

4 (u^{2} - \frac{u}{2} - \frac{u}{2} + \frac{1}{2 2})

Simplify

4 u^{2} - 2 u - 22 u + 2

4 u^{2} - 2 u - 22 u + 2 < 0

Identify the intervals

Find the signs of the factors of

4 u^{2} - 2 u - 22 u + 2

Find the signs of

4 u^{2} - 2 u - 22 u + 2

4 u^{2} - 2 u - 22 u + 2 = 0 : u = \frac{1}{2} or u = \frac{2}{2}

4 u^{2} - 2 u - 22 u + 2 = 0

Write in the standard form

a x^{2} + b x + c = 0

4 u^{2} - (2 + 22) u + 2 = 0

Solve with the quadratic formula

4 u^{2} - (2 + 22) u + 2 = 0

Quadratic Equation Formula:

For

a = 4, b = - 2 - 22, c = 2

u_{1, 2} = \frac{- ( - 2 - 2 2 ) \pm ( - 2 - 2 2 ) ^{2} - 4 \cdot 4 2}{2 \cdot 4}

u_{1, 2} = \frac{- ( - 2 - 2 2 ) \pm ( - 2 - 2 2 ) ^{2} - 4 \cdot 4 2}{2 \cdot 4}

(- 2 - 22)^{2} - 4 \cdot 42 = 22 - 2

(- 2 - 22)^{2} - 4 \cdot 42

Multiply the numbers:

4 \cdot 4 = 16

= (- 2 - 22)^{2} - 162

Expand

(- 2 - 22)^{2} - 162 : 12 - 82

(- 2 - 22)^{2} - 162

(- 2 - 22)^{2} : 12 + 82

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = - 2, b = 22

= (- 2)^{2} - 2 (- 2) \cdot 22 + (22)^{2}

Simplify

(- 2)^{2} - 2 (- 2) \cdot 22 + (22)^{2} : 12 + 82

(- 2)^{2} - 2 (- 2) \cdot 22 + (22)^{2}

Apply rule

- (- a) = a

= (- 2)^{2} + 2 \cdot 2 \cdot 22 + (22)^{2}

(- 2)^{2} = 4

(- 2)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 2^{2}

2^{2} = 4

= 4

2 \cdot 2 \cdot 22 = 82

2 \cdot 2 \cdot 22

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 82

(22)^{2} = 8

(22)^{2}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

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

(2)^{2} : 2

Apply radical rule:

a = a^{\frac{1}{2}}

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2

= 2^{2} \cdot 2

Apply exponent rule:

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

2^{2} \cdot 2 = 2^{2 + 1}

= 2^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2^{3}

2^{3} = 8

= 8

= 4 + 82 + 8

Add the numbers:

4 + 8 = 12

= 12 + 82

= 12 + 82

= 12 + 82 - 162

Add similar elements:

82 - 162 = - 82

= 12 - 82

= 12 - 82

= 4 - 82 + 8

= 2 \cdot 2 - 82 + 8

= (2)^{2} (2)^{2} - 82 + (8)^{2}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= (2)^{2} (2)^{2} - 82 + (22)^{2}

222 \cdot 22 = 82

222 \cdot 22

Multiply the numbers:

2 \cdot 2 = 4

= 4222

Apply exponent rule:

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

222 = 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{\frac{1}{2} + \frac{1}{2} + \frac{1}{2}}

= 4 \cdot 2^{\frac{1}{2} + \frac{1}{2} + \frac{1}{2}}

2^{\frac{1}{2} + \frac{1}{2} + \frac{1}{2}} = 22

2^{\frac{1}{2} + \frac{1}{2} + \frac{1}{2}}

Combine the fractions

\frac{1}{2} + \frac{1}{2} + \frac{1}{2} : \frac{3}{2}

Apply rule

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

= \frac{1 + 1 + 1}{2}

Add the numbers:

1 + 1 + 1 = 3

= \frac{3}{2}

= 2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 22

= 4 \cdot 22

Multiply the numbers:

4 \cdot 2 = 8

= 82

= (22)^{2} - 222 \cdot 22 + (22)^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

(22)^{2} - 222 \cdot 22 + (22)^{2} = (22 - 22)^{2}

= (22 - 22)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(22 - 22)^{2} = (22 - 22)^{2}

= (22 - 22)^{2}

Apply radical rule:

(22 - 22)^{2} = 22 - 22

= 22 - 22

Apply radical rule:

a a = a

22 = 2

= 22 - 2

u_{1, 2} = \frac{- ( - 2 - 2 2 ) \pm ( 2 2 - 2 )}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- ( - 2 - 2 2 ) + 2 2 - 2}{2 \cdot 4}, u_{2} = \frac{- ( - 2 - 2 2 ) - ( 2 2 - 2 )}{2 \cdot 4}

u = \frac{- ( - 2 - 2 2 ) + 2 2 - 2}{2 \cdot 4} : \frac{2}{2}

\frac{- ( - 2 - 2 2 ) + 2 2 - 2}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- ( - 2 - 2 2 ) + 2 2 - 2}{8}

Expand

- (- 2 - 22) + 22 - 2 : 42

- (- 2 - 22) + 22 - 2

- (- 2 - 22) : 2 + 22

- (- 2 - 22)

Distribute parentheses

= - (- 2) - (- 22)

Apply minus-plus rules

- (- a) = a

= 2 + 22

= 2 + 22 + 22 - 2

Simplify

2 + 22 + 22 - 2 : 42

2 + 22 + 22 - 2

Add similar elements:

22 + 22 = 42

= 2 + 42 - 2

2 - 2 = 0

= 42

= 42

= \frac{4 2}{8}

Cancel the common factor:

4

= \frac{2}{2}

u = \frac{- ( - 2 - 2 2 ) - ( 2 2 - 2 )}{2 \cdot 4} : \frac{1}{2}

\frac{- ( - 2 - 2 2 ) - ( 2 2 - 2 )}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- ( - 2 - 2 2 ) - ( 2 2 - 2 )}{8}

Expand

- (- 2 - 22) - (22 - 2) : 4

- (- 2 - 22) - (22 - 2)

- (- 2 - 22) : 2 + 22

- (- 2 - 22)

Distribute parentheses

= - (- 2) - (- 22)

Apply minus-plus rules

- (- a) = a

= 2 + 22

= 2 + 22 - (22 - 2)

- (22 - 2) : - 22 + 2

- (22 - 2)

Distribute parentheses

= - (22) - (- 2)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 22 + 2

= 2 + 22 - 22 + 2

Simplify

2 + 22 - 22 + 2 : 4

2 + 22 - 22 + 2

Add similar elements:

22 - 22 = 0

= 2 + 2

Add the numbers:

2 + 2 = 4

= 4

= 4

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

The solutions to the quadratic equation are:

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

4 u^{2} - 2 u - 22 u + 2 < 0 : \frac{1}{2} < u < \frac{2}{2}

4 u^{2} - 2 u - 22 u + 2 < 0

Factor

4 u^{2} - 2 u - 22 u + 2 : (2 u - 1) (- 2 + 2 u)

4 u^{2} - 2 u - 22 u + 2

= (4 u^{2} - 2 u) + (- 22 u + 2)

Factor out

- 2

from

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

- 22 u + 2

Factor out common term

- 2

= - 2 (2 u - 1)

Factor out

2 u

from

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

4 u^{2} - 2 u

Apply exponent rule:

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

u^{2} = uu

= 4 uu - 2 u

Rewrite

4

2 \cdot 2

= 2 \cdot 2 uu - 2 u

Factor out common term

2 u

= 2 u (2 u - 1)

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

Factor out common term

2 u - 1

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

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

Identify the intervals

Find the signs of the factors of

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

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

Add

1

to 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

Add

1

to 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

Add

1

to 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

- 2 + 2 u

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

- 2 + 2 u = 0

Move

2

to the right side

- 2 + 2 u = 0

Add

2

to both sides

- 2 + 2 u + 2 = 0 + 2

Simplify

2 u = 2

2 u = 2

Divide both sides by

2

2 u = 2

Divide both sides by

2

\frac{2 u}{2} = \frac{2}{2}

Simplify

u = \frac{2}{2}

u = \frac{2}{2}

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

- 2 + 2 u < 0

Move

2

to the right side

- 2 + 2 u < 0

Add

2

to both sides

- 2 + 2 u + 2 < 0 + 2

Simplify

2 u < 2

2 u < 2

Divide both sides by

2

2 u < 2

Divide both sides by

2

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

Simplify

u < \frac{2}{2}

u < \frac{2}{2}

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

- 2 + 2 u > 0

Move

2

to the right side

- 2 + 2 u > 0

Add

2

to both sides

- 2 + 2 u + 2 > 0 + 2

Simplify

2 u > 2

2 u > 2

Divide both sides by

2

2 u > 2

Divide both sides by

2

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

Simplify

u > \frac{2}{2}

u > \frac{2}{2}

Summarize in a table:

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

Identify the intervals that satisfy the required condition:

< 0

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

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

4 u^{2} - 2 u - 22 u + 2 > 0 : u < \frac{1}{2} or u > \frac{2}{2}

4 u^{2} - 2 u - 22 u + 2 > 0

Factor

4 u^{2} - 2 u - 22 u + 2 : (2 u - 1) (- 2 + 2 u)

4 u^{2} - 2 u - 22 u + 2

= (4 u^{2} - 2 u) + (- 22 u + 2)

Factor out

- 2

from

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

- 22 u + 2

Factor out common term

- 2

= - 2 (2 u - 1)

Factor out

2 u

from

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

4 u^{2} - 2 u

Apply exponent rule:

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

u^{2} = uu

= 4 uu - 2 u

Rewrite

4

2 \cdot 2

= 2 \cdot 2 uu - 2 u

Factor out common term

2 u

= 2 u (2 u - 1)

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

Factor out common term

2 u - 1

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

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

Identify the intervals

Find the signs of the factors of

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

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

Add

1

to 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

Add

1

to 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

Add

1

to 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

- 2 + 2 u

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

- 2 + 2 u = 0

Move

2

to the right side

- 2 + 2 u = 0

Add

2

to both sides

- 2 + 2 u + 2 = 0 + 2

Simplify

2 u = 2

2 u = 2

Divide both sides by

2

2 u = 2

Divide both sides by

2

\frac{2 u}{2} = \frac{2}{2}

Simplify

u = \frac{2}{2}

u = \frac{2}{2}

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

- 2 + 2 u < 0

Move

2

to the right side

- 2 + 2 u < 0

Add

2

to both sides

- 2 + 2 u + 2 < 0 + 2

Simplify

2 u < 2

2 u < 2

Divide both sides by

2

2 u < 2

Divide both sides by

2

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

Simplify

u < \frac{2}{2}

u < \frac{2}{2}

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

- 2 + 2 u > 0

Move

2

to the right side

- 2 + 2 u > 0

Add

2

to both sides

- 2 + 2 u + 2 > 0 + 2

Simplify

2 u > 2

2 u > 2

Divide both sides by

2

2 u > 2

Divide both sides by

2

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

Simplify

u > \frac{2}{2}

u > \frac{2}{2}

Summarize in a table:

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

Identify the intervals that satisfy the required condition:

> 0

u < \frac{1}{2} or u > \frac{2}{2}

u < \frac{1}{2} or u > \frac{2}{2}

Summarize in a table:

4 u^{2} - 2 u - 22 u + 2 4 u^{2} - 2 u - 22 u + 2 u < \frac{1}{2} + + u = \frac{1}{2} 00 \frac{1}{2} < u < \frac{2}{2} - - u = \frac{2}{2} 00 u > \frac{2}{2} + +

Identify the intervals that satisfy the required condition:

< 0

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

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

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

Substitute back

u = sin (x)

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

a < u < b

then

a < u and u < b

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

\frac{1}{2} < sin (x) : \frac{π}{6} + 2 πn < x < \frac{5 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 trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

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}

Simplify

π - arcsin (\frac{1}{2}) : \frac{5 π}{6}

π - arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

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}{6}

Simplify

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 = 5 π

= \frac{5 π}{6}

= \frac{5 π}{6}

\frac{π}{6} + 2 πn < x < \frac{5 π}{6} + 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}{6} + 2 πn < x < \frac{5 π}{6} + 2 πn and - \frac{5 π}{4} + 2 πn < x < \frac{π}{4} + 2 πn

Merge Overlapping Intervals

\frac{π}{6} + 2 πn < x < \frac{π}{4} + 2 πn or \frac{3 π}{4} + 2 πn < x < \frac{5 π}{6} + 2 πn

Popular Examples

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cos(2x)<= (sqrt(3))/2

cos (2 x) \leq \frac{3}{2}

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tan (x) < 3