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Popular Trigonometry >

4cos(x+pi/4)>= 2sqrt(2)

Solution

4 cos (x + \frac{π}{4}) \geq 22

Solution

- \frac{π}{2} + 2 πn \leq x \leq 2 πn

Interval Notation

[- \frac{π}{2} + 2 πn, 2 πn]

Decimal

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

Solution steps

4 cos (x + \frac{π}{4}) \geq 22

Divide both sides by

4

4 cos (x + \frac{π}{4}) \geq 22

Divide both sides by

4

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

Simplify

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

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

For

cos (x) \geq a

, if

- 1 < a < 1

then

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

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

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

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

Switch sides

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

Simplify

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

- arccos (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{2}{2}) = \frac{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{π}{4} + 2 πn

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} \geq 0

= x

Simplify

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

- \frac{π}{4} + 2 πn - \frac{π}{4}

Group like terms

= - \frac{π}{4} - \frac{π}{4} + 2 πn

Combine the fractions

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

Apply rule

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

= \frac{- π - π}{4}

Add similar elements:

- π - π = - 2 π

= \frac{- 2 π}{4}

Apply the fraction rule:

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

= - \frac{2 π}{4}

Cancel the common factor:

2

= - \frac{π}{2}

= - \frac{π}{2} + 2 πn

x \geq - \frac{π}{2} + 2 πn

x \geq - \frac{π}{2} + 2 πn

x \geq - \frac{π}{2} + 2 πn

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

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

Simplify

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

arccos (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arccos (\frac{2}{2}) = \frac{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{π}{4} + 2 πn

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

Move

\frac{π}{4}

to the right side

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

Subtract

\frac{π}{4}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} \leq 0

= x

Simplify

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

\frac{π}{4} + 2 πn - \frac{π}{4}

Group like terms

= \frac{π}{4} - \frac{π}{4} + 2 πn

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 2 πn

x \leq 2 πn

x \leq 2 πn

x \leq 2 πn

Combine the intervals

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

Merge Overlapping Intervals

- \frac{π}{2} + 2 πn \leq x \leq 2 πn

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