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cos((3x)/2)cos(x/2)>= 0

Solution

cos (\frac{3 x}{2}) cos (\frac{x}{2}) \geq 0

Solution

x \leq \frac{π}{3} + 2 πn or x = π + 2 πn or x \geq \frac{5 π}{3} + 2 πn

Interval Notation

(- \infty + 2 πn, \frac{π}{3} + 2 πn] \cup x = π + 2 πn \cup [\frac{5 π}{3} + 2 πn, \infty + 2 πn)

Decimal

x \leq 1.04719 \dots + 2 πn or x = 3.14159 \dots + 2 πn or x \geq 5.23598 \dots + 2 πn

Solution steps

cos (\frac{3 x}{2}) cos (\frac{x}{2}) \geq 0

Let:

u = \frac{x}{2}

cos (3 u) cos (u) \geq 0

cos (3 u) cos (u) \geq 0 : πn \leq u \leq \frac{π}{6} + πn or u = \frac{π}{2} + πn or \frac{5 π}{6} + πn \leq u \leq π + πn

cos (3 u) cos (u) \geq 0

Periodicity of

cos (3 u) cos (u) : π

cos (3 u) cos (u)

is composed of the following functions and periods:

cos (3 u)

with periodicity of

\frac{2 π}{3}

The compound periodicity is:

= π

To find the zeroes, set the inequality to zero

cos (3 u) cos (u) = 0

Solve

cos (3 u) cos (u) = 0

for

0 \leq u < π

cos (3 u) cos (u) = 0

Solving each part separately

cos (3 u) = 0 : u = \frac{π}{6} or u = \frac{π}{2} or u = \frac{5 π}{6}

cos (3 u) = 0, 0 \leq u < π

General solutions for

cos (3 u) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

3 u = \frac{π}{2} + 2 πn, 3 u = \frac{3 π}{2} + 2 πn

3 u = \frac{π}{2} + 2 πn, 3 u = \frac{3 π}{2} + 2 πn

Solve

3 u = \frac{π}{2} + 2 πn : u = \frac{π}{6} + \frac{2 πn}{3}

3 u = \frac{π}{2} + 2 πn

Divide both sides by

3

3 u = \frac{π}{2} + 2 πn

Divide both sides by

3

\frac{3 u}{3} = \frac{\frac{π}{2}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 u}{3} = \frac{\frac{π}{2}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Divide the numbers:

\frac{3}{3} = 1

= u

Simplify

\frac{\frac{π}{2}}{3} + \frac{2 πn}{3} : \frac{π}{6} + \frac{2 πn}{3}

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

\frac{\frac{π}{2}}{3} = \frac{π}{6}

\frac{\frac{π}{2}}{3}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π}{6}

= \frac{π}{6} + \frac{2 πn}{3}

u = \frac{π}{6} + \frac{2 πn}{3}

u = \frac{π}{6} + \frac{2 πn}{3}

u = \frac{π}{6} + \frac{2 πn}{3}

Solve

3 u = \frac{3 π}{2} + 2 πn : u = \frac{π}{2} + \frac{2 πn}{3}

3 u = \frac{3 π}{2} + 2 πn

Divide both sides by

3

3 u = \frac{3 π}{2} + 2 πn

Divide both sides by

3

\frac{3 u}{3} = \frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 u}{3} = \frac{\frac{3 π}{2}}{3} + \frac{2 πn}{3}

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Divide the numbers:

\frac{3}{3} = 1

= u

Simplify

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

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

\frac{\frac{3 π}{2}}{3} = \frac{π}{2}

\frac{\frac{3 π}{2}}{3}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{3 π}{6}

Cancel the common factor:

3

= \frac{π}{2}

= \frac{π}{2} + \frac{2 πn}{3}

u = \frac{π}{2} + \frac{2 πn}{3}

u = \frac{π}{2} + \frac{2 πn}{3}

u = \frac{π}{2} + \frac{2 πn}{3}

u = \frac{π}{6} + \frac{2 πn}{3}, u = \frac{π}{2} + \frac{2 πn}{3}

Solutions for the range

0 \leq u < π

u = \frac{π}{6}, u = \frac{π}{2}, u = \frac{5 π}{6}

cos (u) = 0 : u = \frac{π}{2}

cos (u) = 0, 0 \leq u < π

General solutions for

cos (u) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn

Solutions for the range

0 \leq u < π

u = \frac{π}{2}

Combine all the solutions

\frac{π}{6} or \frac{π}{2} or \frac{5 π}{6}

The intervals between the zeros

0 < u < \frac{π}{6}, \frac{π}{6} < u < \frac{π}{2}, \frac{π}{2} < u < \frac{5 π}{6}, \frac{5 π}{6} < u < π

Summarize in a table:

cos (3 u) cos (u) cos (3 u) cos (u) u = 0 + + + 0 < u < \frac{π}{6} + + + u = \frac{π}{6} 0 + 0 \frac{π}{6} < u < \frac{π}{2} - + - u = \frac{π}{2} 000 \frac{π}{2} < u < \frac{5 π}{6} + - - u = \frac{5 π}{6} 0 - 0 \frac{5 π}{6} < u < π - - + u = π - - +

Identify the intervals that satisfy the required condition:

\geq 0

u = 0 or 0 < u < \frac{π}{6} or u = \frac{π}{6} or u = \frac{π}{2} or u = \frac{5 π}{6} or \frac{5 π}{6} < u < π or u = π

Merge Overlapping Intervals

0 \leq u \leq \frac{π}{6} or u = \frac{π}{2} or \frac{5 π}{6} \leq u < π or u = π