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Popular >

cos(x)<= sin(x)

Solution

cos (x) \leq sin (x)

Solution

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

Interval Notation

[\frac{π}{4} + 2 πn, \frac{5 π}{4} + 2 πn]

Decimal Notation

0.78539 \dots + 2 πn \leq x \leq 3.92699 \dots + 2 πn

Solution steps

cos (x) \leq sin (x)

Move

sin (x)

to the left side

cos (x) \leq sin (x)

Subtract

sin (x)

from both sides

cos (x) - sin (x) \leq sin (x) - sin (x)

cos (x) - sin (x) \leq 0

cos (x) - sin (x) \leq 0

Use the following identity:

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

2 cos (\frac{π}{4} + x) \leq 0

Divide both sides by

2

2 cos (\frac{π}{4} + x) \leq 0

Divide both sides by

2

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

Simplify

cos (\frac{π}{4} + x) \leq 0

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

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

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

Switch sides

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

\frac{π}{4} + x - \frac{π}{4} \geq \frac{π}{2} + 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} \geq 0

= 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}

x \geq 2 πn + \frac{π}{4}

x \geq 2 πn + \frac{π}{4}

x \geq 2 πn + \frac{π}{4}

\frac{π}{4} + x \leq 2 π - arccos (0) + 2 πn : x \leq \frac{5 π}{4} + 2 πn

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

\frac{π}{4} + x - \frac{π}{4} \leq 2 π - \frac{π}{2} + 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} \leq 0

= x

Simplify

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

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

Group like terms

= 2 π + 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 π + 2 πn - \frac{3 π}{4}

x \leq 2 π + 2 πn - \frac{3 π}{4}

x \leq 2 π + 2 πn - \frac{3 π}{4}

x \leq 2 π + 2 πn - \frac{3 π}{4}

Simplify

2 π - \frac{3 π}{4} : \frac{5 π}{4}

2 π - \frac{3 π}{4}

Convert element to fraction:

2 π = \frac{2 π 4}{4}

= \frac{2 π 4}{4} - \frac{3 π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π 4 - 3 π}{4}

2 π 4 - 3 π = 5 π

2 π 4 - 3 π

Multiply the numbers:

2 \cdot 4 = 8

= 8 π - 3 π

Add similar elements:

8 π - 3 π = 5 π

= 5 π

= \frac{5 π}{4}

x \leq \frac{5 π}{4} + 2 πn

Combine the intervals

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

Merge Overlapping Intervals

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

Popular Examples

2sin(t)>0

2 sin (t) > 0

1-tan(x)<= 0

1 - tan (x) \leq 0

sin(x+23)cos(x-37)>(sqrt(3))/2

sin (x + 2 3^{\circ}) cos (x - 3 7^{\circ}) > \frac{3}{2}

cos(2x)>= (sqrt(3))/2

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

cos(x)<-1

cos (x) < - 1