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Popular Trigonometry >

sin(x)>cos(x)

Solution

sin (x) > cos (x)

Solution

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

Interval Notation

(\frac{π}{4} + 2 πn, \frac{5 π}{4} + 2 πn)

Decimal

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

Solution steps

sin (x) > cos (x)

Move

cos (x)

to the left side

sin (x) > cos (x)

Subtract

cos (x)

from both sides

sin (x) - cos (x) > cos (x) - cos (x)

sin (x) - cos (x) > 0

sin (x) - cos (x) > 0

Use the following identity:

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

- 2 cos (\frac{π}{4} + x) > 0

Multiply both sides by

- 1

- 2 cos (\frac{π}{4} + x) > 0

Multiply both sides by -1 (reverse the inequality)

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

Simplify

2 cos (\frac{π}{4} + x) < 0

2 cos (\frac{π}{4} + x) < 0

Divide both sides by

2

2 cos (\frac{π}{4} + x) < 0

Divide both sides by

2

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

Simplify

cos (\frac{π}{4} + x) < 0

cos (\frac{π}{4} + x) < 0

For

cos (x) < a

, if

- 1 < a \leq 1

then

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

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

a < u < b

then

a < u and u < b

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

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

arccos (0) + 2 πn < \frac{π}{4} + x

Switch sides

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

Move

\frac{π}{4}

to the right side

\frac{π}{4} + x > \frac{π}{2} + 2 πn

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

x > 2 πn + \frac{π}{4}

x > 2 πn + \frac{π}{4}

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

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

x < 2 π + 2 πn - \frac{3 π}{4}

x < 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 < \frac{5 π}{4} + 2 πn

Combine the intervals

x > 2 πn + \frac{π}{4} and x < \frac{5 π}{4} + 2 πn

Merge Overlapping Intervals

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

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