English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

2sin^2(x)-1<= 0

Solution

2 sin^{2} (x) - 1 \leq 0

Solution

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

Interval Notation

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

Decimal

2 πn \leq x \leq 0.78539 \dots + 2 πn or 2.35619 \dots + 2 πn \leq x \leq 3.92699 \dots + 2 πn or 5.49778 \dots + 2 πn \leq x < 6.28318 \dots + 2 πn

Solution steps

2 sin^{2} (x) - 1 \leq 0

Move

1

to the right side

2 sin^{2} (x) - 1 \leq 0

Add

1

to both sides

2 sin^{2} (x) - 1 + 1 \leq 0 + 1

Simplify

2 sin^{2} (x) \leq 1

2 sin^{2} (x) \leq 1

Divide both sides by

2

2 sin^{2} (x) \leq 1

Divide both sides by

2

\frac{2 sin ^{2} ( x )}{2} \leq \frac{1}{2}

Simplify

sin^{2} (x) \leq \frac{1}{2}

sin^{2} (x) \leq \frac{1}{2}

For

u^{n} \leq a

, if

n

is even then

- \frac{1}{2} \leq sin (x) \leq \frac{1}{2}

a \leq u \leq b

then

a \leq u and u \leq b

- \frac{1}{2} \leq sin (x) and sin (x) \leq \frac{1}{2}

- \frac{1}{2} \leq sin (x) : - \frac{π}{4} + 2 πn \leq x \leq \frac{5 π}{4} + 2 πn

- \frac{1}{2} \leq sin (x)

Switch sides

sin (x) \geq - \frac{1}{2}

For

sin (x) \geq a

, if

- 1 < a < 1

then

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

arcsin (- \frac{1}{2}) + 2 πn \leq x \leq π - arcsin (- \frac{1}{2}) + 2 πn

Simplify

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

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

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{2}) = - arcsin (\frac{1}{2})

= - arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{4}

arcsin (\frac{1}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{4}

= - \frac{π}{4}

Simplify

π - arcsin (- \frac{1}{2}) : \frac{5 π}{4}

π - arcsin (- \frac{1}{2})

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

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

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{2}) = - arcsin (\frac{1}{2})

= - arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{4}

arcsin (\frac{1}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{4}

= - \frac{π}{4}

= π - (- \frac{π}{4})

Simplify

π - (- \frac{π}{4})

Apply rule

- (- a) = a

= π + \frac{π}{4}

Convert element to fraction:

π = \frac{π 4}{4}

= \frac{π 4}{4} + \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 + π}{4}

Add similar elements:

4 π + π = 5 π

= \frac{5 π}{4}

= \frac{5 π}{4}

- \frac{π}{4} + 2 πn \leq x \leq \frac{5 π}{4} + 2 πn

sin (x) \leq \frac{1}{2} : - \frac{5 π}{4} + 2 πn \leq x \leq \frac{π}{4} + 2 πn

sin (x) \leq \frac{1}{2}

For

sin (x) \leq a

, if

- 1 < a < 1

then

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

- π - arcsin (\frac{1}{2}) + 2 πn \leq x \leq arcsin (\frac{1}{2}) + 2 πn

Simplify

- π - arcsin (\frac{1}{2}) : - \frac{5 π}{4}

- π - arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{4}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= - π - \frac{π}{4}

Simplify

- π - \frac{π}{4}

Convert element to fraction:

π = \frac{π 4}{4}

= - \frac{π 4}{4} - \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 4 - π}{4}

Add similar elements:

- 4 π - π = - 5 π

= \frac{- 5 π}{4}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{5 π}{4}

= - \frac{5 π}{4}

Simplify

arcsin (\frac{1}{2}) : \frac{π}{4}

arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{4}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{4}

- \frac{5 π}{4} + 2 πn \leq x \leq \frac{π}{4} + 2 πn

Combine the intervals

- \frac{π}{4} + 2 πn \leq x \leq \frac{5 π}{4} + 2 πn and - \frac{5 π}{4} + 2 πn \leq x \leq \frac{π}{4} + 2 πn

Merge Overlapping Intervals

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

Popular Examples

2sin^2(x)+3sin(x)>= 2

cos(x)<=-1/2

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

tan(θ)>0,sin(θ)<0

(1+2sin(x))/((cos(2x)))>0