English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sin(2x)<cos(2x)

Solution

sin (2 x) < cos (2 x)

Solution

- \frac{3 π}{8} + πn < x < \frac{π}{8} + πn

Interval Notation

(- \frac{3 π}{8} + πn, \frac{π}{8} + πn)

Decimal Notation

- 1.17809 \dots + πn < x < 0.39269 \dots + πn

Solution steps

sin (2 x) < cos (2 x)

Move

cos (2 x)

to the left side

sin (2 x) < cos (2 x)

Subtract

cos (2 x)

from both sides

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

sin (2 x) - cos (2 x) < 0

sin (2 x) - cos (2 x) < 0

Use the following identity:

- cos (x) + sin (x) = - 2 cos (\frac{π}{4} + x)

- 2 cos (\frac{π}{4} + 2 x) < 0

Multiply both sides by

- 1

- 2 cos (\frac{π}{4} + 2 x) < 0

Multiply both sides by -1 (reverse the inequality)

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

Simplify

2 cos (\frac{π}{4} + 2 x) > 0

2 cos (\frac{π}{4} + 2 x) > 0

Divide both sides by

2

2 cos (\frac{π}{4} + 2 x) > 0

Divide both sides by

2

\frac{2 cos ( \frac{π}{4} + 2 x )}{2} > \frac{0}{2}

Simplify

cos (\frac{π}{4} + 2 x) > 0

cos (\frac{π}{4} + 2 x) > 0

For

cos (x) > a

, if

- 1 \leq a < 1

then

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

- arccos (0) + 2 πn < (\frac{π}{4} + 2 x) < arccos (0) + 2 πn

a < u < b

then

a < u and u < b

- arccos (0) + 2 πn < \frac{π}{4} + 2 x and \frac{π}{4} + 2 x < arccos (0) + 2 πn

- arccos (0) + 2 πn < \frac{π}{4} + 2 x : x > πn - \frac{3 π}{8}

- arccos (0) + 2 πn < \frac{π}{4} + 2 x

Switch sides

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

\frac{π}{4} + 2 x - \frac{π}{4} : 2 x

\frac{π}{4} + 2 x - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} > 0

= 2 x

Simplify

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

- \frac{π}{2} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{2} - \frac{π}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

\frac{π}{2} = \frac{π 2}{2 \cdot 2} = \frac{π 2}{4}

= - \frac{π 2}{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 2 - π}{4}

Add similar elements:

- 2 π - π = - 3 π

= \frac{- 3 π}{4}

Apply the fraction rule:

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

= 2 πn - \frac{3 π}{4}

2 x > 2 πn - \frac{3 π}{4}

2 x > 2 πn - \frac{3 π}{4}

2 x > 2 πn - \frac{3 π}{4}

Divide both sides by

2

2 x > 2 πn - \frac{3 π}{4}

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} - \frac{\frac{3 π}{4}}{2} : πn - \frac{3 π}{8}

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

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{3 π}{4}}{2} = \frac{3 π}{8}

\frac{\frac{3 π}{4}}{2}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{3 π}{8}

= πn - \frac{3 π}{8}

x > πn - \frac{3 π}{8}

x > πn - \frac{3 π}{8}

x > πn - \frac{3 π}{8}

\frac{π}{4} + 2 x < arccos (0) + 2 πn : x < πn + \frac{π}{8}

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

\frac{π}{4} + 2 x - \frac{π}{4} : 2 x

\frac{π}{4} + 2 x - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} < 0

= 2 x

Simplify

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

\frac{π}{2} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{2} - \frac{π}{4}

Least Common Multiplier of

2, 4 : 4

2, 4

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

2

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

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

4

For

\frac{π}{2} :

multiply the denominator and numerator by

2

\frac{π}{2} = \frac{π 2}{2 \cdot 2} = \frac{π 2}{4}

= \frac{π 2}{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 2 - π}{4}

Add similar elements:

2 π - π = π

= 2 πn + \frac{π}{4}

2 x < 2 πn + \frac{π}{4}

2 x < 2 πn + \frac{π}{4}

2 x < 2 πn + \frac{π}{4}

Divide both sides by

2

2 x < 2 πn + \frac{π}{4}

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} + \frac{\frac{π}{4}}{2} : πn + \frac{π}{8}

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

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π}{8}

= πn + \frac{π}{8}

x < πn + \frac{π}{8}

x < πn + \frac{π}{8}

x < πn + \frac{π}{8}

Combine the intervals

x > πn - \frac{3 π}{8} and x < πn + \frac{π}{8}

Merge Overlapping Intervals

- \frac{3 π}{8} + πn < x < \frac{π}{8} + πn

Popular Examples

pi/2-arctan(x^4)>0.0001

\frac{π}{2} - arctan (x^{4}) > 0.0001

2cos^2(x)>1

2 cos^{2} (x) > 1

sin(2x)>= cos(x)

sin (2 x) \geq cos (x)

3sin(2x)>0

3 sin (2 x) > 0

cos(x)-sin(x)>=-3/2

cos (x) - sin (x) \geq - \frac{3}{2}