English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

1-2cos^2(1/2 x)>0

Solution

1 - 2 cos^{2} (\frac{1}{2} x) > 0

Solution

\frac{π}{2} + 4 πn < x < \frac{3 π}{2} + 4 πn or \frac{5 π}{2} + 4 πn < x < \frac{7 π}{2} + 4 πn

Interval Notation

(\frac{π}{2} + 4 πn, \frac{3 π}{2} + 4 πn) \cup (\frac{5 π}{2} + 4 πn, \frac{7 π}{2} + 4 πn)

Decimal

1.57079 \dots + 4 πn < x < 4.71238 \dots + 4 πn or 7.85398 \dots + 4 πn < x < 10.99557 \dots + 4 πn

Solution steps

1 - 2 cos^{2} (\frac{1}{2} x) > 0

Move

1

to the right side

1 - 2 cos^{2} (\frac{1}{2} x) > 0

Subtract

1

from both sides

1 - 2 cos^{2} (\frac{1}{2} x) - 1 > 0 - 1

Simplify

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

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

Multiply both sides by

- 1

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

Multiply both sides by -1 (reverse the inequality)

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

Simplify

2 cos^{2} (\frac{1}{2} x) < 1

2 cos^{2} (\frac{1}{2} x) < 1

Divide both sides by

2

2 cos^{2} (\frac{1}{2} x) < 1

Divide both sides by

2

\frac{2 cos ^{2} ( \frac{1}{2} x )}{2} < \frac{1}{2}

Simplify

cos^{2} (\frac{1}{2} x) < \frac{1}{2}

cos^{2} (\frac{1}{2} x) < \frac{1}{2}

For

u^{n} < a

, if

n

is even then

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

a < u < b

then

a < u and u < b

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

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

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

Switch sides

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

For

cos (x) > a

, if

- 1 \leq a < 1

then

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

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

a < u < b

then

a < u and u < b

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

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

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

Switch sides

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

Simplify

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

- arccos (- \frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{1}{2}) = \frac{3 π}{4}

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{3 π}{4} + 2 πn

\frac{1}{2} x > - \frac{3 π}{4} + 2 πn

Multiply both sides by

2

\frac{1}{2} x > - \frac{3 π}{4} + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x > - 2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x > - 2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

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

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

- 2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn : - \frac{3 π}{2} + 4 πn

- 2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

Multiply fractions:

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

= \frac{3 π 2}{4}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= - \frac{3 π}{2} + 4 πn

x > - \frac{3 π}{2} + 4 πn

x > - \frac{3 π}{2} + 4 πn

x > - \frac{3 π}{2} + 4 πn

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

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

Simplify

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

arccos (- \frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{1}{2}) = \frac{3 π}{4}

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{3 π}{4} + 2 πn

\frac{1}{2} x < \frac{3 π}{4} + 2 πn

Multiply both sides by

2

\frac{1}{2} x < \frac{3 π}{4} + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x < 2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x < 2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

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

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn : \frac{3 π}{2} + 4 πn

2 \cdot \frac{3 π}{4} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{4} = \frac{3 π}{2}

2 \cdot \frac{3 π}{4}

Multiply fractions:

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

= \frac{3 π 2}{4}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 π}{4}

Cancel the common factor:

2

= \frac{3 π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{3 π}{2} + 4 πn

x < \frac{3 π}{2} + 4 πn

x < \frac{3 π}{2} + 4 πn

x < \frac{3 π}{2} + 4 πn

Combine the intervals

x > - \frac{3 π}{2} + 4 πn and x < \frac{3 π}{2} + 4 πn

Merge Overlapping Intervals

- \frac{3 π}{2} + 4 πn < x < \frac{3 π}{2} + 4 πn

cos (\frac{1}{2} x) < \frac{1}{2} : \frac{π}{2} + 4 πn < x < \frac{7 π}{2} + 4 πn

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

a < u < b

then

a < u and u < b

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

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

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

Switch sides

\frac{1}{2} x > arccos (\frac{1}{2}) + 2 πn

Simplify

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

arccos (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{π}{4}

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}{4} + 2 πn

\frac{1}{2} x > \frac{π}{4} + 2 πn

Multiply both sides by

2

\frac{1}{2} x > \frac{π}{4} + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x > 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x > 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

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

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

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

2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{2} + 4 πn

x > \frac{π}{2} + 4 πn

x > \frac{π}{2} + 4 πn

x > \frac{π}{2} + 4 πn

\frac{1}{2} x < 2 π - arccos (\frac{1}{2}) + 2 πn : x < \frac{7 π}{2} + 4 πn

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

Simplify

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

2 π - arccos (\frac{1}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{π}{4}

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{π}{4} + 2 πn

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

Multiply both sides by

2

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

Multiply both sides by

2

2 \cdot \frac{1}{2} x < 2 \cdot 2 π - 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x < 2 \cdot 2 π - 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

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

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

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

2 \cdot 2 π - 2 \cdot \frac{π}{4} + 2 \cdot 2 πn

2 \cdot 2 π = 4 π

2 \cdot 2 π

Multiply the numbers:

2 \cdot 2 = 4

= 4 π

2 \cdot \frac{π}{4} = \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 4 π - \frac{π}{2} + 4 πn

x < 4 π - \frac{π}{2} + 4 πn

x < 4 π - \frac{π}{2} + 4 πn

Simplify

4 π - \frac{π}{2} : \frac{7 π}{2}

4 π - \frac{π}{2}

Convert element to fraction:

4 π = \frac{4 π 2}{2}

= \frac{4 π 2}{2} - \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{4 π 2 - π}{2}

4 π 2 - π = 7 π

4 π 2 - π

Multiply the numbers:

4 \cdot 2 = 8

= 8 π - π

Add similar elements:

8 π - π = 7 π

= 7 π

= \frac{7 π}{2}

x < \frac{7 π}{2} + 4 πn

x < \frac{7 π}{2} + 4 πn

Combine the intervals

x > \frac{π}{2} + 4 πn and x < \frac{7 π}{2} + 4 πn

Merge Overlapping Intervals

\frac{π}{2} + 4 πn < x < \frac{7 π}{2} + 4 πn

Combine the intervals

- \frac{3 π}{2} + 4 πn < x < \frac{3 π}{2} + 4 πn and \frac{π}{2} + 4 πn < x < \frac{7 π}{2} + 4 πn

Merge Overlapping Intervals

\frac{π}{2} + 4 πn < x < \frac{3 π}{2} + 4 πn or \frac{5 π}{2} + 4 πn < x < \frac{7 π}{2} + 4 πn

Popular Examples

sqrt(3)cos(4x)+sin(4x)>sqrt(2)

sin(5x)>5,0<= x<= 2pi

-cos(x)<=-sin(2x)

cos(y)>-1

2cos(x)+cos^2(x)>0