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5-8cos(x)+4cos(2x)<0

Solution

5 - 8 cos (x) + 4 cos (2 x) < 0

Solution

arccos (\frac{2 + 2}{4}) + 2 πn < x < arccos (\frac{- 2 + 2}{4}) + 2 πn or - arccos (\frac{- 2 + 2}{4}) + 2 π + 2 πn < x < 2 π - arccos (\frac{2 + 2}{4}) + 2 πn

Interval Notation

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

Decimal Notation

0.54802 \dots + 2 πn < x < 1.42382 \dots + 2 πn or 4.85936 \dots + 2 πn < x < 5.73515 \dots + 2 πn

Solution steps

5 - 8 cos (x) + 4 cos (2 x) < 0

Use the following identity:

cos (2 x) = - 1 + 2 cos^{2} (x)

5 + 4 (- 1 + 2 cos^{2} (x)) - 8 cos (x) < 0

Simplify

5 + 4 (- 1 + 2 cos^{2} (x)) - 8 cos (x) : 8 cos^{2} (x) - 8 cos (x) + 1

5 + 4 (- 1 + 2 cos^{2} (x)) - 8 cos (x)

Expand

4 (- 1 + 2 cos^{2} (x)) : - 4 + 8 cos^{2} (x)

4 (- 1 + 2 cos^{2} (x))

Apply the distributive law:

a (b + c) = ab + a c

a = 4, b = - 1, c = 2 cos^{2} (x)

= 4 (- 1) + 4 \cdot 2 cos^{2} (x)

Apply minus-plus rules

+ (- a) = - a

= - 4 \cdot 1 + 4 \cdot 2 cos^{2} (x)

Simplify

- 4 \cdot 1 + 4 \cdot 2 cos^{2} (x) : - 4 + 8 cos^{2} (x)

- 4 \cdot 1 + 4 \cdot 2 cos^{2} (x)

Multiply the numbers:

4 \cdot 1 = 4

= - 4 + 4 \cdot 2 cos^{2} (x)

Multiply the numbers:

4 \cdot 2 = 8

= - 4 + 8 cos^{2} (x)

= - 4 + 8 cos^{2} (x)

= 5 - 4 + 8 cos^{2} (x) - 8 cos (x)

Subtract the numbers:

5 - 4 = 1

= 8 cos^{2} (x) - 8 cos (x) + 1

8 cos^{2} (x) - 8 cos (x) + 1 < 0

Let:

u = cos (x)

8 u^{2} - 8 u + 1 < 0

8 u^{2} - 8 u + 1 < 0 : - \frac{2}{4} + \frac{1}{2} < u < \frac{2}{4} + \frac{1}{2}

8 u^{2} - 8 u + 1 < 0

Complete the square

8 u^{2} - 8 u + 1 : 8 (u - \frac{1}{2})^{2} - 1

8 u^{2} - 8 u + 1

Write

8 u^{2} - 8 u + 1

in the form:

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

Factor out

8

8 (u^{2} - u + \frac{1}{8})

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

2 a = - 1

Divide both sides by

2

2 a = - 1

Divide both sides by

2

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

Simplify

a = - \frac{1}{2}

a = - \frac{1}{2}

Add and subtract

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

8 (u^{2} - u + \frac{1}{8} + (- \frac{1}{2})^{2} - (- \frac{1}{2})^{2})

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

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

8 ((u - \frac{1}{2})^{2} + \frac{1}{8} - (- \frac{1}{2})^{2})

Simplify

8 (u - \frac{1}{2})^{2} - 1

8 (u - \frac{1}{2})^{2} - 1 < 0

Move

1

to the right side

8 (u - \frac{1}{2})^{2} - 1 < 0

Add

1

to both sides

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

Simplify

8 (u - \frac{1}{2})^{2} < 1

8 (u - \frac{1}{2})^{2} < 1

Divide both sides by

8

8 (u - \frac{1}{2})^{2} < 1

Divide both sides by

8

\frac{8 ( u - \frac{1}{2} ) ^{2}}{8} < \frac{1}{8}

Simplify

(u - \frac{1}{2})^{2} < \frac{1}{8}

(u - \frac{1}{2})^{2} < \frac{1}{8}

For

u^{n} < a

, if

n

is even then

- \frac{1}{8} < u - \frac{1}{2} < \frac{1}{8}

a < u < b

then

a < u and u < b

- \frac{1}{8} < u - \frac{1}{2} and u - \frac{1}{2} < \frac{1}{8}

- \frac{1}{8} < u - \frac{1}{2} : u > - \frac{2}{4} + \frac{1}{2}

- \frac{1}{8} < u - \frac{1}{2}

Switch sides

u - \frac{1}{2} > - \frac{1}{8}

Simplify

- \frac{1}{8} : - \frac{1}{2 2}

- \frac{1}{8}

Simplify

\frac{1}{8} : \frac{1}{2 2}

\frac{1}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{8}

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

= \frac{1}{2 2}

= - \frac{1}{2 2}

Apply rule

1 = 1

= - \frac{1}{2 2}

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

Move

\frac{1}{2}

to the right side

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

Add

\frac{1}{2}

to both sides

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

Simplify

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

Simplify

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

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

Add similar elements:

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

= u

Simplify

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

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

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

\frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

222 = 4

222

Apply exponent rule:

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

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

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

Add similar elements:

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

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

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

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

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

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

u > - \frac{2}{4} + \frac{1}{2}

u > - \frac{2}{4} + \frac{1}{2}

u > - \frac{2}{4} + \frac{1}{2}

u - \frac{1}{2} < \frac{1}{8} : u < \frac{2}{4} + \frac{1}{2}

u - \frac{1}{2} < \frac{1}{8}

Simplify

\frac{1}{8} : \frac{1}{2 2}

\frac{1}{8}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{8}

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

= \frac{1}{2 2}

Apply rule

1 = 1

= \frac{1}{2 2}

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

Move

\frac{1}{2}

to the right side

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

Add

\frac{1}{2}

to both sides

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

Simplify

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

Simplify

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

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

Add similar elements:

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

= u

Simplify

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

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

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

\frac{1}{2 2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

222 = 4

222

Apply exponent rule:

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

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

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

Add similar elements:

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

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

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

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

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2}{4}

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

u < \frac{2}{4} + \frac{1}{2}

u < \frac{2}{4} + \frac{1}{2}

u < \frac{2}{4} + \frac{1}{2}

Combine the intervals

u > - \frac{2}{4} + \frac{1}{2} and u < \frac{2}{4} + \frac{1}{2}

Merge Overlapping Intervals

u > - \frac{2}{4} + \frac{1}{2} and u < \frac{2}{4} + \frac{1}{2}

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

u > - \frac{2}{4} + \frac{1}{2}

and

u < \frac{2}{4} + \frac{1}{2}

- \frac{2}{4} + \frac{1}{2} < u < \frac{2}{4} + \frac{1}{2}

- \frac{2}{4} + \frac{1}{2} < u < \frac{2}{4} + \frac{1}{2}

- \frac{2}{4} + \frac{1}{2} < u < \frac{2}{4} + \frac{1}{2}

Substitute back

u = cos (x)

- \frac{2}{4} + \frac{1}{2} < cos (x) < \frac{2}{4} + \frac{1}{2}

a < u < b

then

a < u and u < b

- \frac{2}{4} + \frac{1}{2} < cos (x) and cos (x) < \frac{2}{4} + \frac{1}{2}

- \frac{2}{4} + \frac{1}{2} < cos (x) : - arccos (\frac{- 2 + 2}{4}) + 2 πn < x < arccos (\frac{- 2 + 2}{4}) + 2 πn

- \frac{2}{4} + \frac{1}{2} < cos (x)

Switch sides

cos (x) > - \frac{2}{4} + \frac{1}{2}

For

cos (x) > a

, if

- 1 \leq a < 1

then

- arccos (a) + 2 πn < x < arccos (a) + 2 πn

- arccos (- \frac{2}{4} + \frac{1}{2}) + 2 πn < x < arccos (- \frac{2}{4} + \frac{1}{2}) + 2 πn

Simplify

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

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

Join

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

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

Least Common Multiplier of

4, 2 : 4

4, 2

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

4

2

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{2} :

multiply the denominator and numerator by

2

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

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

Since the denominators are equal, combine the fractions:

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

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

Factor

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

- 2 + 2

2 = 22

= - 2 + 22

Factor out common term

2

= 2 (- 1 + 2)

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

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

\frac{2 ( - 1 + 2 )}{2 ^{2}} : \frac{- 1 + 2}{2 ^{\frac{3}{2}}}

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

Apply radical rule:

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

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

Subtract the numbers:

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

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

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

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

= - arccos (\frac{2 - 2}{4})

Simplify

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

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

Join

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

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

Least Common Multiplier of

4, 2 : 4

4, 2

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

4

2

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{2} :

multiply the denominator and numerator by

2

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

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

Since the denominators are equal, combine the fractions:

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

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

Factor

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

- 2 + 2

2 = 22

= - 2 + 22

Factor out common term

2

= 2 (- 1 + 2)

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

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

\frac{2 ( - 1 + 2 )}{2 ^{2}} : \frac{- 1 + 2}{2 ^{\frac{3}{2}}}

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

Apply radical rule:

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

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

Subtract the numbers:

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

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

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

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

= arccos (\frac{- 2 + 2}{4})

- arccos (\frac{- 2 + 2}{4}) + 2 πn < x < arccos (\frac{- 2 + 2}{4}) + 2 πn

cos (x) < \frac{2}{4} + \frac{1}{2} : arccos (\frac{2 + 2}{4}) + 2 πn < x < 2 π - arccos (\frac{2 + 2}{4}) + 2 πn

cos (x) < \frac{2}{4} + \frac{1}{2}

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

arccos (\frac{2}{4} + \frac{1}{2}) + 2 πn < x < 2 π - arccos (\frac{2}{4} + \frac{1}{2}) + 2 πn

Simplify

arccos (\frac{2}{4} + \frac{1}{2}) : arccos (\frac{2 + 2}{4})

arccos (\frac{2}{4} + \frac{1}{2})

Join

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

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

Least Common Multiplier of

4, 2 : 4

4, 2

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

4

2

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{2} :

multiply the denominator and numerator by

2

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

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

Since the denominators are equal, combine the fractions:

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

= \frac{2 + 2}{4}

Factor

2 + 2 : 2 (1 + 2)

2 + 2

2 = 22

= 2 + 22

Factor out common term

2

= 2 (1 + 2)

= \frac{2 ( 1 + 2 )}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

\frac{2 ( 1 + 2 )}{2 ^{2}} : \frac{1 + 2}{2 ^{\frac{3}{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 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

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

Subtract the numbers:

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

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

= \frac{1 + 2}{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 2}

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

= arccos (\frac{2 + 2}{4})

Simplify

2 π - arccos (\frac{2}{4} + \frac{1}{2}) : 2 π - arccos (\frac{2 + 2}{4})

2 π - arccos (\frac{2}{4} + \frac{1}{2})

Join

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

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

Least Common Multiplier of

4, 2 : 4

4, 2

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Multiply each factor the greatest number of times it occurs in either

4

2

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{1}{2} :

multiply the denominator and numerator by

2

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

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

Since the denominators are equal, combine the fractions:

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

= \frac{2 + 2}{4}

= 2 π - arccos (\frac{2 + 2}{4})

arccos (\frac{2 + 2}{4}) + 2 πn < x < 2 π - arccos (\frac{2 + 2}{4}) + 2 πn

Combine the intervals

- arccos (\frac{- 2 + 2}{4}) + 2 πn < x < arccos (\frac{- 2 + 2}{4}) + 2 πn and arccos (\frac{2 + 2}{4}) + 2 πn < x < 2 π - arccos (\frac{2 + 2}{4}) + 2 πn

Merge Overlapping Intervals

arccos (\frac{2 + 2}{4}) + 2 πn < x < arccos (\frac{- 2 + 2}{4}) + 2 πn or - arccos (\frac{- 2 + 2}{4}) + 2 π + 2 πn < x < 2 π - arccos (\frac{2 + 2}{4}) + 2 πn

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