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Popular Trigonometry >

4(cos(x))^2>1

Solution

4 (cos (x))^{2} > 1

Solution

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

Interval Notation

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

Decimal

- 1.04719 \dots + 2 πn < x < 1.04719 \dots + 2 πn or 2.09439 \dots + 2 πn < x < 4.18879 \dots + 2 πn

Solution steps

4 (cos (x))^{2} > 1

Divide both sides by

4

4 cos^{2} (x) > 1

Divide both sides by

4

\frac{4 cos ^{2} ( x )}{4} > \frac{1}{4}

Simplify

cos^{2} (x) > \frac{1}{4}

cos^{2} (x) > \frac{1}{4}

For

u^{n} > a

, if

n

is even then

cos (x) < - \frac{1}{4} or cos (x) > \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

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

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

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

Simplify

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

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

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}

Simplify

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

2 π - arccos (- \frac{1}{2})

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}

Simplify

2 π - \frac{2 π}{3}

Convert element to fraction:

2 π = \frac{2 π 3}{3}

= \frac{2 π 3}{3} - \frac{2 π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π 3 - 2 π}{3}

2 π 3 - 2 π = 4 π

2 π 3 - 2 π

Multiply the numbers:

2 \cdot 3 = 6

= 6 π - 2 π

Add similar elements:

6 π - 2 π = 4 π

= 4 π

= \frac{4 π}{3}

= \frac{4 π}{3}

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

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

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

Simplify

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

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

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{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{π}{3}

Simplify

arccos (\frac{1}{2}) : \frac{π}{3}

arccos (\frac{1}{2})

Use the following trivial identity:

arccos (\frac{1}{2}) = \frac{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{π}{3}

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

Combine the intervals

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

Merge Overlapping Intervals

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

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