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Popular Trigonometry >

cos(x/2)<= 0

Solution

cos (\frac{x}{2}) \leq 0

Solution

π + 4 πn \leq x \leq 3 π + 4 πn

Interval Notation

[π + 4 πn, 3 π + 4 πn]

Decimal

3.14159 \dots + 4 πn \leq x \leq 9.42477 \dots + 4 πn

Solution steps

cos (\frac{x}{2}) \leq 0

For

cos (x) \leq a

, if

- 1 < a < 1

then

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

arccos (0) + 2 πn \leq \frac{x}{2} \leq 2 π - arccos (0) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

arccos (0) + 2 πn \leq \frac{x}{2} and \frac{x}{2} \leq 2 π - arccos (0) + 2 πn

arccos (0) + 2 πn \leq \frac{x}{2} : x \geq π + 4 πn

arccos (0) + 2 πn \leq \frac{x}{2}

Switch sides

\frac{x}{2} \geq arccos (0) + 2 πn

Simplify

arccos (0) + 2 πn : \frac{π}{2} + 2 πn

arccos (0) + 2 πn

Use the following trivial identity:

arccos (0) = \frac{π}{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}{2} + 2 πn

\frac{x}{2} \geq \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{x}{2} \geq \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

x \geq π + 4 πn

x \geq π + 4 πn

x \geq π + 4 πn

\frac{x}{2} \leq 2 π - arccos (0) + 2 πn : x \leq 3 π + 4 πn

\frac{x}{2} \leq 2 π - arccos (0) + 2 πn

Simplify

2 π - arccos (0) + 2 πn : 2 π - \frac{π}{2} + 2 πn

2 π - arccos (0) + 2 πn

Use the following trivial identity:

arccos (0) = \frac{π}{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}

= 2 π - \frac{π}{2} + 2 πn

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

Multiply both sides by

2

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

Multiply both sides by

2

\frac{2 x}{2} \leq 2 \cdot 2 π - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} \leq 2 \cdot 2 π - 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

2 \cdot 2 π - 2 \cdot \frac{π}{2} + 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}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 4 π - π + 4 πn

Add similar elements:

4 π - π = 3 π

= 3 π + 4 πn

x \leq 3 π + 4 πn

x \leq 3 π + 4 πn

x \leq 3 π + 4 πn

Combine the intervals

x \geq π + 4 πn and x \leq 3 π + 4 πn

Merge Overlapping Intervals

π + 4 πn \leq x \leq 3 π + 4 πn

Popular Examples

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

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

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

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

cos(y)>-1