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cos(4/3 x+pi/6)>= 1

Solution

cos (\frac{4}{3} x + \frac{π}{6}) \geq 1

Solution

x = - \frac{π}{8} + \frac{3 π}{2} n

Decimal Notation

x = - 0.39269 \dots + \frac{3 π}{2} n

Solution steps

cos (\frac{4}{3} x + \frac{π}{6}) \geq 1

For

cos (x) \geq a

, if

- 1 < a < 1

then

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

- arccos (1) + 2 πn \leq (\frac{4}{3} x + \frac{π}{6}) \leq arccos (1) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

- arccos (1) + 2 πn \leq \frac{4}{3} x + \frac{π}{6} and \frac{4}{3} x + \frac{π}{6} \leq arccos (1) + 2 πn

- arccos (1) + 2 πn \leq \frac{4}{3} x + \frac{π}{6} : x \geq \frac{3 πn}{2} - \frac{π}{8}

- arccos (1) + 2 πn \leq \frac{4}{3} x + \frac{π}{6}

Switch sides

\frac{4}{3} x + \frac{π}{6} \geq - arccos (1) + 2 πn

Simplify

- arccos (1) + 2 πn : 2 πn

- arccos (1) + 2 πn

Use the following trivial identity:

arccos (1) = 0

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}

= - 0 + 2 πn

- 0 + 2 πn = 2 πn

= 2 πn

\frac{4}{3} x + \frac{π}{6} \geq 2 πn

Move

\frac{π}{6}

to the right side

\frac{4}{3} x + \frac{π}{6} \geq 2 πn

Subtract

\frac{π}{6}

from both sides

\frac{4}{3} x + \frac{π}{6} - \frac{π}{6} \geq 2 πn - \frac{π}{6}

Simplify

\frac{4}{3} x \geq 2 πn - \frac{π}{6}

\frac{4}{3} x \geq 2 πn - \frac{π}{6}

Multiply both sides by

3

\frac{4}{3} x \geq 2 πn - \frac{π}{6}

Multiply both sides by

3

3 \cdot \frac{4}{3} x \geq 3 \cdot 2 πn - 3 \cdot \frac{π}{6}

Simplify

3 \cdot \frac{4}{3} x \geq 3 \cdot 2 πn - 3 \cdot \frac{π}{6}

Simplify

3 \cdot \frac{4}{3} x : 4 x

3 \cdot \frac{4}{3} x

Multiply fractions:

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

= \frac{4 \cdot 3}{3} x

Cancel the common factor:

3

= x \cdot 4

Simplify

3 \cdot 2 πn - 3 \cdot \frac{π}{6} : 6 πn - \frac{π}{2}

3 \cdot 2 πn - 3 \cdot \frac{π}{6}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

3 \cdot \frac{π}{6} = \frac{π}{2}

3 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 3}{6}

Cancel the common factor:

3

= \frac{π}{2}

= 6 πn - \frac{π}{2}

4 x \geq 6 πn - \frac{π}{2}

4 x \geq 6 πn - \frac{π}{2}

4 x \geq 6 πn - \frac{π}{2}

Divide both sides by

4

4 x \geq 6 πn - \frac{π}{2}

Divide both sides by

4

\frac{4 x}{4} \geq \frac{6 πn}{4} - \frac{\frac{π}{2}}{4}

Simplify

\frac{4 x}{4} \geq \frac{6 πn}{4} - \frac{\frac{π}{2}}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{6 πn}{4} - \frac{\frac{π}{2}}{4} : \frac{3 πn}{2} - \frac{π}{8}

\frac{6 πn}{4} - \frac{\frac{π}{2}}{4}

\frac{6 πn}{4} = \frac{3 πn}{2}

\frac{6 πn}{4}

Cancel the common factor:

2

= \frac{3 πn}{2}

\frac{\frac{π}{2}}{4} = \frac{π}{8}

\frac{\frac{π}{2}}{4}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π}{8}

= \frac{3 πn}{2} - \frac{π}{8}

x \geq \frac{3 πn}{2} - \frac{π}{8}

x \geq \frac{3 πn}{2} - \frac{π}{8}

x \geq \frac{3 πn}{2} - \frac{π}{8}

\frac{4}{3} x + \frac{π}{6} \leq arccos (1) + 2 πn : x \leq \frac{3 πn}{2} - \frac{π}{8}

\frac{4}{3} x + \frac{π}{6} \leq arccos (1) + 2 πn

Simplify

arccos (1) + 2 πn : 2 πn

arccos (1) + 2 πn

Use the following trivial identity:

arccos (1) = 0

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}

= 0 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

\frac{4}{3} x + \frac{π}{6} \leq 2 πn

Move

\frac{π}{6}

to the right side

\frac{4}{3} x + \frac{π}{6} \leq 2 πn

Subtract

\frac{π}{6}

from both sides

\frac{4}{3} x + \frac{π}{6} - \frac{π}{6} \leq 2 πn - \frac{π}{6}

Simplify

\frac{4}{3} x \leq 2 πn - \frac{π}{6}

\frac{4}{3} x \leq 2 πn - \frac{π}{6}

Multiply both sides by

3

\frac{4}{3} x \leq 2 πn - \frac{π}{6}

Multiply both sides by

3

3 \cdot \frac{4}{3} x \leq 3 \cdot 2 πn - 3 \cdot \frac{π}{6}

Simplify

3 \cdot \frac{4}{3} x \leq 3 \cdot 2 πn - 3 \cdot \frac{π}{6}

Simplify

3 \cdot \frac{4}{3} x : 4 x

3 \cdot \frac{4}{3} x

Multiply fractions:

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

= \frac{4 \cdot 3}{3} x

Cancel the common factor:

3

= x \cdot 4

Simplify

3 \cdot 2 πn - 3 \cdot \frac{π}{6} : 6 πn - \frac{π}{2}

3 \cdot 2 πn - 3 \cdot \frac{π}{6}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

3 \cdot \frac{π}{6} = \frac{π}{2}

3 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 3}{6}

Cancel the common factor:

3

= \frac{π}{2}

= 6 πn - \frac{π}{2}

4 x \leq 6 πn - \frac{π}{2}

4 x \leq 6 πn - \frac{π}{2}

4 x \leq 6 πn - \frac{π}{2}

Divide both sides by

4

4 x \leq 6 πn - \frac{π}{2}

Divide both sides by

4

\frac{4 x}{4} \leq \frac{6 πn}{4} - \frac{\frac{π}{2}}{4}

Simplify

\frac{4 x}{4} \leq \frac{6 πn}{4} - \frac{\frac{π}{2}}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{6 πn}{4} - \frac{\frac{π}{2}}{4} : \frac{3 πn}{2} - \frac{π}{8}

\frac{6 πn}{4} - \frac{\frac{π}{2}}{4}

\frac{6 πn}{4} = \frac{3 πn}{2}

\frac{6 πn}{4}

Cancel the common factor:

2

= \frac{3 πn}{2}

\frac{\frac{π}{2}}{4} = \frac{π}{8}

\frac{\frac{π}{2}}{4}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{π}{8}

= \frac{3 πn}{2} - \frac{π}{8}

x \leq \frac{3 πn}{2} - \frac{π}{8}

x \leq \frac{3 πn}{2} - \frac{π}{8}

x \leq \frac{3 πn}{2} - \frac{π}{8}

Combine the intervals

x \geq \frac{3 πn}{2} - \frac{π}{8} and x \leq \frac{3 πn}{2} - \frac{π}{8}

Merge Overlapping Intervals

x = - \frac{π}{8} + \frac{3 π}{2} n

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