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Popular >

cos(pix)<-1/2

Solution

cos (π x) < - \frac{1}{2}

Solution

\frac{2}{3} + 2 n < x < \frac{4}{3} + 2 n

Interval Notation

(\frac{2}{3} + 2 n, \frac{4}{3} + 2 n)

Decimal Notation

0.66666 \dots + 2 n < x < 1.33333 \dots + 2 n

Solution steps

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

a < u < b

then

a < u and u < b

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

arccos (- \frac{1}{2}) + 2 πn < π x : x > \frac{2}{3} + 2 n

arccos (- \frac{1}{2}) + 2 πn < π x

Switch sides

π x > 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

π x > \frac{2 π}{3} + 2 πn

Divide both sides by

π

π x > \frac{2 π}{3} + 2 πn

Divide both sides by

π

\frac{π x}{π} > \frac{\frac{2 π}{3}}{π} + \frac{2 πn}{π}

Simplify

\frac{π x}{π} > \frac{\frac{2 π}{3}}{π} + \frac{2 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{2 π}{3}}{π} + \frac{2 πn}{π} : \frac{2}{3} + 2 n

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

\frac{\frac{2 π}{3}}{π} = \frac{2}{3}

\frac{\frac{2 π}{3}}{π}

Apply the fraction rule:

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

= \frac{2 π}{3 π}

Cancel the common factor:

π

= \frac{2}{3}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= \frac{2}{3} + 2 n

x > \frac{2}{3} + 2 n

x > \frac{2}{3} + 2 n

x > \frac{2}{3} + 2 n

π x < 2 π - arccos (- \frac{1}{2}) + 2 πn : x < \frac{4}{3} + 2 n

π x < 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

π x < 2 π - \frac{2 π}{3} + 2 πn

Divide both sides by

π

π x < 2 π - \frac{2 π}{3} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

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

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

\frac{2 π}{π} = 2

\frac{2 π}{π}

Cancel the common factor:

π

= 2

\frac{\frac{2 π}{3}}{π} = \frac{2}{3}

\frac{\frac{2 π}{3}}{π}

Apply the fraction rule:

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

= \frac{2 π}{3 π}

Cancel the common factor:

π

= \frac{2}{3}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= 2 - \frac{2}{3} + 2 n

x < 2 - \frac{2}{3} + 2 n

x < 2 - \frac{2}{3} + 2 n

Simplify

2 - \frac{2}{3} : \frac{4}{3}

2 - \frac{2}{3}

Convert element to fraction:

2 = \frac{2 \cdot 3}{3}

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

Since the denominators are equal, combine the fractions:

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

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

2 \cdot 3 - 2 = 4

2 \cdot 3 - 2

Multiply the numbers:

2 \cdot 3 = 6

= 6 - 2

Subtract the numbers:

6 - 2 = 4

= 4

= \frac{4}{3}

x < \frac{4}{3} + 2 n

x < \frac{4}{3} + 2 n

Combine the intervals

x > \frac{2}{3} + 2 n and x < \frac{4}{3} + 2 n

Merge Overlapping Intervals

\frac{2}{3} + 2 n < x < \frac{4}{3} + 2 n

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