English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin(pi/4-x)< 1/2

Solution

sin (\frac{π}{4} - x) < \frac{1}{2}

Solution

\frac{π}{12} + 2 πn < x < \frac{17 π}{12} + 2 πn

Interval Notation

(\frac{π}{12} + 2 πn, \frac{17 π}{12} + 2 πn)

Decimal Notation

0.26179 \dots + 2 πn < x < 4.45058 \dots + 2 πn

Solution steps

sin (\frac{π}{4} - x) < \frac{1}{2}

Factor out

- 1

from

\frac{π}{4} - x : - (- \frac{π}{4} + x)

sin (- (- \frac{π}{4} + x)) < \frac{1}{2}

Use the following identity:

sin (- x) = - sin (x)

- sin (- \frac{π}{4} + x) < \frac{1}{2}

Multiply both sides by

- 1

- sin (- \frac{π}{4} + x) < \frac{1}{2}

Multiply both sides by -1 (reverse the inequality)

(- sin (- \frac{π}{4} + x)) (- 1) > \frac{1 \cdot ( - 1 )}{2}

Simplify

sin (- \frac{π}{4} + x) > - \frac{1}{2}

sin (- \frac{π}{4} + x) > - \frac{1}{2}

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

a < u < b

then

a < u and u < b

arcsin (- \frac{1}{2}) + 2 πn < - \frac{π}{4} + x and - \frac{π}{4} + x < π - arcsin (- \frac{1}{2}) + 2 πn

arcsin (- \frac{1}{2}) + 2 πn < - \frac{π}{4} + x : x > 2 πn + \frac{π}{12}

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

Switch sides

- \frac{π}{4} + x > arcsin (- \frac{1}{2}) + 2 πn

Simplify

arcsin (- \frac{1}{2}) + 2 πn : - \frac{π}{6} + 2 πn

arcsin (- \frac{1}{2}) + 2 πn

arcsin (- \frac{1}{2}) = - \frac{π}{6}

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}{6}

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}{6}

= - \frac{π}{6}

= - \frac{π}{6} + 2 πn

- \frac{π}{4} + x > - \frac{π}{6} + 2 πn

Move

\frac{π}{4}

to the right side

- \frac{π}{4} + x > - \frac{π}{6} + 2 πn

Add

\frac{π}{4}

to both sides

- \frac{π}{4} + x + \frac{π}{4} > - \frac{π}{6} + 2 πn + \frac{π}{4}

Simplify

- \frac{π}{4} + x + \frac{π}{4} > - \frac{π}{6} + 2 πn + \frac{π}{4}

Simplify

- \frac{π}{4} + x + \frac{π}{4} : x

- \frac{π}{4} + x + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} > 0

= x

Simplify

- \frac{π}{6} + 2 πn + \frac{π}{4} : 2 πn + \frac{π}{12}

- \frac{π}{6} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{6} + \frac{π}{4}

Least Common Multiplier of

6, 4 : 12

6, 4

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

6

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

= - \frac{π 2}{12} + \frac{π 3}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 2 + π 3}{12}

Add similar elements:

- 2 π + 3 π = π

= 2 πn + \frac{π}{12}

x > 2 πn + \frac{π}{12}

x > 2 πn + \frac{π}{12}

x > 2 πn + \frac{π}{12}

- \frac{π}{4} + x < π - arcsin (- \frac{1}{2}) + 2 πn : x < \frac{17 π}{12} + 2 πn

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

Simplify

π - arcsin (- \frac{1}{2}) + 2 πn : π + \frac{π}{6} + 2 πn

π - arcsin (- \frac{1}{2}) + 2 πn

arcsin (- \frac{1}{2}) = - \frac{π}{6}

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}{6}

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}{6}

= - \frac{π}{6}

= π - (- \frac{π}{6}) + 2 πn

Apply rule

- (- a) = a

= π + \frac{π}{6} + 2 πn

- \frac{π}{4} + x < π + \frac{π}{6} + 2 πn

Move

\frac{π}{4}

to the right side

- \frac{π}{4} + x < π + \frac{π}{6} + 2 πn

Add

\frac{π}{4}

to both sides

- \frac{π}{4} + x + \frac{π}{4} < π + \frac{π}{6} + 2 πn + \frac{π}{4}

Simplify

- \frac{π}{4} + x + \frac{π}{4} < π + \frac{π}{6} + 2 πn + \frac{π}{4}

Simplify

- \frac{π}{4} + x + \frac{π}{4} : x

- \frac{π}{4} + x + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} < 0

= x

Simplify

π + \frac{π}{6} + 2 πn + \frac{π}{4} : π + 2 πn + \frac{5 π}{12}

π + \frac{π}{6} + 2 πn + \frac{π}{4}

Group like terms

= π + 2 πn + \frac{π}{6} + \frac{π}{4}

Least Common Multiplier of

6, 4 : 12

6, 4

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

6

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

= \frac{π 2}{12} + \frac{π 3}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + π 3}{12}

Add similar elements:

2 π + 3 π = 5 π

= π + 2 πn + \frac{5 π}{12}

x < π + 2 πn + \frac{5 π}{12}

x < π + 2 πn + \frac{5 π}{12}

x < π + 2 πn + \frac{5 π}{12}

Simplify

π + \frac{5 π}{12} : \frac{17 π}{12}

π + \frac{5 π}{12}

Convert element to fraction:

π = \frac{π 12}{12}

= \frac{π 12}{12} + \frac{5 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 12 + 5 π}{12}

Add similar elements:

12 π + 5 π = 17 π

= \frac{17 π}{12}

x < \frac{17 π}{12} + 2 πn

Combine the intervals

x > 2 πn + \frac{π}{12} and x < \frac{17 π}{12} + 2 πn

Merge Overlapping Intervals

\frac{π}{12} + 2 πn < x < \frac{17 π}{12} + 2 πn

Popular Examples

cos^2(x)>14

cos^{2} (x) > 14

cos(y)>=-1

cos (y) \geq - 1

3cos(3 x/2-pi/4)-1>0

3 cos (3 \frac{x}{2} - \frac{π}{4}) - 1 > 0

2sin(x)<= 1,2cos(x)<sqrt(3)

2 sin (x) \leq 1, 2 cos (x) < 3

solvefor θ,3r^2cos^2(θ)+r^2sin^2(θ)<= 1 solve for

θ, 3 r^{2} cos^{2} (θ) + r^{2} sin^{2} (θ) \leq 1