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Popular Trigonometry >

sin(x)-cos(x)>= 1

Solution

sin (x) - cos (x) \geq 1

Solution

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

Interval Notation

[\frac{π}{2} + 2 πn, π + 2 πn]

Decimal

1.57079 \dots + 2 πn \leq x \leq 3.14159 \dots + 2 πn

Solution steps

sin (x) - cos (x) \geq 1

Use the following identity:

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

- 2 cos (\frac{π}{4} + x) \geq 1

Multiply both sides by

- 1

- 2 cos (\frac{π}{4} + x) \geq 1

Multiply both sides by -1 (reverse the inequality)

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

Simplify

2 cos (\frac{π}{4} + x) \leq - 1

2 cos (\frac{π}{4} + x) \leq - 1

Divide both sides by

2

2 cos (\frac{π}{4} + x) \leq - 1

Divide both sides by

2

\frac{2 cos ( \frac{π}{4} + x )}{2} \leq \frac{- 1}{2}

Simplify

\frac{2 cos ( \frac{π}{4} + x )}{2} \leq \frac{- 1}{2}

Simplify

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

\frac{2 cos ( \frac{π}{4} + x )}{2}

Cancel the common factor:

2

= cos (\frac{π}{4} + x)

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

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

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

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

For

cos (x) \leq a

, if

- 1 < a < 1

then

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

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

a \leq u \leq b

then

a \leq u and u \leq b

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

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

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

Switch sides

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

Simplify

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

arccos (- \frac{2}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{2}{2}) = \frac{3 π}{4}

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{3 π}{4} + 2 πn

\frac{π}{4} + x \geq \frac{3 π}{4} + 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} + x \geq \frac{3 π}{4} + 2 πn

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

\frac{3 π}{4} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{3 π}{4}

Combine the fractions

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

Apply rule

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

= \frac{- π + 3 π}{4}

Add similar elements:

- π + 3 π = 2 π

= \frac{2 π}{4}

Cancel the common factor:

2

= \frac{π}{2}

= 2 πn + \frac{π}{2}

x \geq 2 πn + \frac{π}{2}

x \geq 2 πn + \frac{π}{2}

x \geq 2 πn + \frac{π}{2}

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

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

Simplify

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

2 π - arccos (- \frac{2}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{2}{2}) = \frac{3 π}{4}

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{3 π}{4} + 2 πn

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

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

Group like terms

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

Combine the fractions

- \frac{π}{4} - \frac{3 π}{4} : - π

Apply rule

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

= \frac{- π - 3 π}{4}

Add similar elements:

- π - 3 π = - 4 π

= \frac{- 4 π}{4}

Apply the fraction rule:

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

= - \frac{4 π}{4}

Divide the numbers:

\frac{4}{4} = 1

= - π

= 2 π + 2 πn - π

Group like terms

= 2 π - π + 2 πn

Add similar elements:

2 π - π = π

= π + 2 πn

x \leq π + 2 πn

x \leq π + 2 πn

x \leq π + 2 πn

Combine the intervals

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

Merge Overlapping Intervals

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

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