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Popular >

cos(2x)>2cos(2x)-0.5

Solution

cos (2 x) > 2 cos (2 x) - 0.5

Solution

\frac{π}{6} + πn < x < \frac{5 π}{6} + πn

Interval Notation

(\frac{π}{6} + πn, \frac{5 π}{6} + πn)

Decimal Notation

0.52359 \dots + πn < x < 2.61799 \dots + πn

Solution steps

cos (2 x) > 2 cos (2 x) - 0.5

Let:

u = cos (2 x)

u > 2 u - 0.5

u > 2 u - 0.5 : u < 0.5

u > 2 u - 0.5

Move

2 u

to the left side

u > 2 u - 0.5

Subtract

2 u

from both sides

u - 2 u > 2 u - 0.5 - 2 u

Simplify

- u > - 0.5

- u > - 0.5

Multiply both sides by

- 1

- u > - 0.5

Multiply both sides by -1 (reverse the inequality)

(- u) (- 1) < (- 0.5) (- 1)

Simplify

u < 0.5

u < 0.5

u < 0.5

Substitute back

u = cos (2 x)

cos (2 x) < 0.5

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

arccos (0.5) + 2 πn < 2 x < 2 π - arccos (0.5) + 2 πn

a < u < b

then

a < u and u < b

arccos (0.5) + 2 πn < 2 x and 2 x < 2 π - arccos (0.5) + 2 πn

arccos (0.5) + 2 πn < 2 x : x > \frac{π}{6} + πn

arccos (0.5) + 2 πn < 2 x

Switch sides

2 x > arccos (0.5) + 2 πn

Simplify

arccos (0.5) + 2 πn : \frac{π}{3} + 2 πn

arccos (0.5) + 2 πn

arccos (0.5) = \frac{π}{3}

arccos (0.5)

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

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{π}{3}

arccos (\frac{1}{2})

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{π}{3}

= \frac{π}{3}

= \frac{π}{3} + 2 πn

2 x > \frac{π}{3} + 2 πn

Divide both sides by

2

2 x > \frac{π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{6} + πn

x > \frac{π}{6} + πn

x > \frac{π}{6} + πn

x > \frac{π}{6} + πn

2 x < 2 π - arccos (0.5) + 2 πn : x < \frac{5 π}{6} + πn

2 x < 2 π - arccos (0.5) + 2 πn

Simplify

2 π - arccos (0.5) + 2 πn : 2 π - \frac{π}{3} + 2 πn

2 π - arccos (0.5) + 2 πn

arccos (0.5) = \frac{π}{3}

arccos (0.5)

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

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{π}{3}

arccos (\frac{1}{2})

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{π}{3}

= \frac{π}{3}

= 2 π - \frac{π}{3} + 2 πn

2 x < 2 π - \frac{π}{3} + 2 πn

Divide both sides by

2

2 x < 2 π - \frac{π}{3} + 2 πn

Divide both sides by

2

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

Simplify

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

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

\frac{2 π}{2} = π

\frac{2 π}{2}

Divide the numbers:

\frac{2}{2} = 1

= π

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= π - \frac{π}{6} + πn

x < π - \frac{π}{6} + πn

x < π - \frac{π}{6} + πn

Simplify

π - \frac{π}{6} : \frac{5 π}{6}

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}

x < \frac{5 π}{6} + πn

x < \frac{5 π}{6} + πn

Combine the intervals

x > \frac{π}{6} + πn and x < \frac{5 π}{6} + πn

Merge Overlapping Intervals

\frac{π}{6} + πn < x < \frac{5 π}{6} + πn

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