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Popular Trigonometry >

cos(2x+pi/6)<=-1/2

Solution

cos (2 x + \frac{π}{6}) \leq - \frac{1}{2}

Solution

\frac{π}{4} + πn \leq x \leq \frac{7 π}{12} + πn

Interval Notation

[\frac{π}{4} + πn, \frac{7 π}{12} + πn]

Decimal

0.78539 \dots + πn \leq x \leq 1.83259 \dots + πn

Solution steps

cos (2 x + \frac{π}{6}) \leq - \frac{1}{2}

For

cos (x) \leq a

, if

- 1 < a < 1

then

arccos (a) + 2 πn \leq x \leq 2 π - arccos (a) + 2 πn

arccos (- \frac{1}{2}) + 2 πn \leq (2 x + \frac{π}{6}) \leq 2 π - arccos (- \frac{1}{2}) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

arccos (- \frac{1}{2}) + 2 πn \leq 2 x + \frac{π}{6} and 2 x + \frac{π}{6} \leq 2 π - arccos (- \frac{1}{2}) + 2 πn

arccos (- \frac{1}{2}) + 2 πn \leq 2 x + \frac{π}{6} : x \geq πn + \frac{π}{4}

arccos (- \frac{1}{2}) + 2 πn \leq 2 x + \frac{π}{6}

Switch sides

2 x + \frac{π}{6} \geq arccos (- \frac{1}{2}) + 2 πn

Simplify

arccos (- \frac{1}{2}) + 2 πn : \frac{2 π}{3} + 2 πn

arccos (- \frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{1}{2}) = \frac{2 π}{3}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{2 π}{3} + 2 πn

2 x + \frac{π}{6} \geq \frac{2 π}{3} + 2 πn

Move

\frac{π}{6}

to the right side

2 x + \frac{π}{6} \geq \frac{2 π}{3} + 2 πn

Subtract

\frac{π}{6}

from both sides

2 x + \frac{π}{6} - \frac{π}{6} \geq \frac{2 π}{3} + 2 πn - \frac{π}{6}

Simplify

2 x + \frac{π}{6} - \frac{π}{6} \geq \frac{2 π}{3} + 2 πn - \frac{π}{6}

Simplify

2 x + \frac{π}{6} - \frac{π}{6} : 2 x

2 x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} \geq 0

= 2 x

Simplify

\frac{2 π}{3} + 2 πn - \frac{π}{6} : 2 πn + \frac{π}{2}

\frac{2 π}{3} + 2 πn - \frac{π}{6}

Group like terms

= 2 πn - \frac{π}{6} + \frac{2 π}{3}

Least Common Multiplier of

6, 3 : 6

6, 3

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

6

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{2 π}{3} :

multiply the denominator and numerator by

2

\frac{2 π}{3} = \frac{2 π 2}{3 \cdot 2} = \frac{4 π}{6}

= - \frac{π}{6} + \frac{4 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π + 4 π}{6}

Add similar elements:

- π + 4 π = 3 π

= \frac{3 π}{6}

Cancel the common factor:

3

= 2 πn + \frac{π}{2}

2 x \geq 2 πn + \frac{π}{2}

2 x \geq 2 πn + \frac{π}{2}

2 x \geq 2 πn + \frac{π}{2}

Divide both sides by

2

2 x \geq 2 πn + \frac{π}{2}

Divide both sides by

2

\frac{2 x}{2} \geq \frac{2 πn}{2} + \frac{\frac{π}{2}}{2}

Simplify

\frac{2 x}{2} \geq \frac{2 πn}{2} + \frac{\frac{π}{2}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} + \frac{\frac{π}{2}}{2} : πn + \frac{π}{4}

\frac{2 πn}{2} + \frac{\frac{π}{2}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

= πn + \frac{π}{4}

x \geq πn + \frac{π}{4}

x \geq πn + \frac{π}{4}

x \geq πn + \frac{π}{4}

2 x + \frac{π}{6} \leq 2 π - arccos (- \frac{1}{2}) + 2 πn : x \leq \frac{7 π}{12} + πn

2 x + \frac{π}{6} \leq 2 π - arccos (- \frac{1}{2}) + 2 πn

Simplify

2 π - arccos (- \frac{1}{2}) + 2 πn : 2 π - \frac{2 π}{3} + 2 πn

2 π - arccos (- \frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{1}{2}) = \frac{2 π}{3}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{2 π}{3} + 2 πn

2 x + \frac{π}{6} \leq 2 π - \frac{2 π}{3} + 2 πn

Move

\frac{π}{6}

to the right side

2 x + \frac{π}{6} \leq 2 π - \frac{2 π}{3} + 2 πn

Subtract

\frac{π}{6}

from both sides

2 x + \frac{π}{6} - \frac{π}{6} \leq 2 π - \frac{2 π}{3} + 2 πn - \frac{π}{6}

Simplify

2 x + \frac{π}{6} - \frac{π}{6} \leq 2 π - \frac{2 π}{3} + 2 πn - \frac{π}{6}

Simplify

2 x + \frac{π}{6} - \frac{π}{6} : 2 x

2 x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} \leq 0

= 2 x

Simplify

2 π - \frac{2 π}{3} + 2 πn - \frac{π}{6} : 2 π + 2 πn - \frac{5 π}{6}

2 π - \frac{2 π}{3} + 2 πn - \frac{π}{6}

Group like terms

= 2 π + 2 πn - \frac{π}{6} - \frac{2 π}{3}

Least Common Multiplier of

6, 3 : 6

6, 3

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

6

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{2 π}{3} :

multiply the denominator and numerator by

2

\frac{2 π}{3} = \frac{2 π 2}{3 \cdot 2} = \frac{4 π}{6}

= - \frac{π}{6} - \frac{4 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π - 4 π}{6}

Add similar elements:

- π - 4 π = - 5 π

= \frac{- 5 π}{6}

Apply the fraction rule:

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

= 2 π + 2 πn - \frac{5 π}{6}

2 x \leq 2 π + 2 πn - \frac{5 π}{6}

2 x \leq 2 π + 2 πn - \frac{5 π}{6}

2 x \leq 2 π + 2 πn - \frac{5 π}{6}

Divide both sides by

2

2 x \leq 2 π + 2 πn - \frac{5 π}{6}

Divide both sides by

2

\frac{2 x}{2} \leq \frac{2 π}{2} + \frac{2 πn}{2} - \frac{\frac{5 π}{6}}{2}

Simplify

\frac{2 x}{2} \leq \frac{2 π}{2} + \frac{2 πn}{2} - \frac{\frac{5 π}{6}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 π}{2} + \frac{2 πn}{2} - \frac{\frac{5 π}{6}}{2} : π + πn - \frac{5 π}{12}

\frac{2 π}{2} + \frac{2 πn}{2} - \frac{\frac{5 π}{6}}{2}

\frac{2 π}{2} = π

\frac{2 π}{2}

Divide the numbers:

\frac{2}{2} = 1

= π

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

Apply the fraction rule:

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

= \frac{5 π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{5 π}{12}

= π + πn - \frac{5 π}{12}

x \leq π + πn - \frac{5 π}{12}

x \leq π + πn - \frac{5 π}{12}

Simplify

π - \frac{5 π}{12} : \frac{7 π}{12}

π - \frac{5 π}{12}

Convert element to fraction:

π = \frac{π 12}{12}

= \frac{π 12}{12} - \frac{5 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 12 - 5 π}{12}

Add similar elements:

12 π - 5 π = 7 π

= \frac{7 π}{12}

x \leq \frac{7 π}{12} + πn

x \leq \frac{7 π}{12} + πn

Combine the intervals

x \geq πn + \frac{π}{4} and x \leq \frac{7 π}{12} + πn

Merge Overlapping Intervals

\frac{π}{4} + πn \leq x \leq \frac{7 π}{12} + πn

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