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(cos(5x)+cos(3x))/(cos(4x))<= 0

Solution

\frac{cos ( 5 x ) + cos ( 3 x )}{cos ( 4 x )} \leq 0

Solution

\frac{π}{2} + 2 πn \leq x < \frac{5 π}{8} + 2 πn or \frac{5 π}{8} + 2 πn < x < \frac{7 π}{8} + 2 πn or \frac{7 π}{8} + 2 πn < x < \frac{9 π}{8} + 2 πn or \frac{9 π}{8} + 2 πn < x < \frac{11 π}{8} + 2 πn or \frac{11 π}{8} + 2 πn < x \leq \frac{3 π}{2} + 2 πn

Interval Notation

[\frac{π}{2} + 2 πn, \frac{5 π}{8} + 2 πn) \cup (\frac{5 π}{8} + 2 πn, \frac{7 π}{8} + 2 πn) \cup (\frac{7 π}{8} + 2 πn, \frac{9 π}{8} + 2 πn) \cup (\frac{9 π}{8} + 2 πn, \frac{11 π}{8} + 2 πn) \cup (\frac{11 π}{8} + 2 πn, \frac{3 π}{2} + 2 πn]

Decimal Notation

1.57079 \dots + 2 πn \leq x < 1.96349 \dots + 2 πn or 1.96349 \dots + 2 πn < x < 2.74889 \dots + 2 πn or 2.74889 \dots + 2 πn < x < 3.53429 \dots + 2 πn or 3.53429 \dots + 2 πn < x < 4.31968 \dots + 2 πn or 4.31968 \dots + 2 πn < x \leq 4.71238 \dots + 2 πn

Solution steps

\frac{cos ( 5 x ) + cos ( 3 x )}{cos ( 4 x )} \leq 0

Periodicity of

\frac{cos ( 5 x ) + cos ( 3 x )}{cos ( 4 x )} : 2 π

\frac{cos ( 5 x ) + cos ( 3 x )}{cos ( 4 x )}

is composed of the following functions and periods:

cos (5 x)

with periodicity of

\frac{2 π}{5}

The compound periodicity is:

= 2 π

Find the zeroes and undifined points of

\frac{cos ( 5 x ) + cos ( 3 x )}{cos ( 4 x )}

for

0 \leq x < 2 π

To find the zeroes, set the inequality to zero

\frac{cos ( 5 x ) + cos ( 3 x )}{cos ( 4 x )} = 0

\frac{cos ( 5 x ) + cos ( 3 x )}{cos ( 4 x )} = 0, 0 \leq x < 2 π : x = \frac{π}{2}, x = \frac{3 π}{2}

\frac{cos ( 5 x ) + cos ( 3 x )}{cos ( 4 x )} = 0, 0 \leq x < 2 π

Rewrite using trig identities

\frac{cos ( 5 x ) + cos ( 3 x )}{cos ( 4 x )}

Use the Sum to Product identity:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

= \frac{2 cos ( \frac{5 x + 3 x}{2} ) cos ( \frac{5 x - 3 x}{2} )}{cos ( 4 x )}

Simplify

\frac{2 cos ( \frac{5 x + 3 x}{2} ) cos ( \frac{5 x - 3 x}{2} )}{cos ( 4 x )} : 2 cos (x)

\frac{2 cos ( \frac{5 x + 3 x}{2} ) cos ( \frac{5 x - 3 x}{2} )}{cos ( 4 x )}

2 cos (\frac{5 x + 3 x}{2}) cos (\frac{5 x - 3 x}{2}) = 2 cos (\frac{8 x}{2}) cos (\frac{2 x}{2})

2 cos (\frac{5 x + 3 x}{2}) cos (\frac{5 x - 3 x}{2})

Add similar elements:

5 x + 3 x = 8 x

= 2 cos (\frac{8 x}{2}) cos (\frac{5 x - 3 x}{2})

Add similar elements:

5 x - 3 x = 2 x

= 2 cos (\frac{8 x}{2}) cos (\frac{2 x}{2})

= \frac{2 cos ( \frac{8 x}{2} ) cos ( \frac{2 x}{2} )}{cos ( 4 x )}

Divide the numbers:

\frac{8}{2} = 4

= \frac{2 cos ( 4 x ) cos ( \frac{2 x}{2} )}{cos ( 4 x )}

Divide the numbers:

\frac{2}{2} = 1

= \frac{2 cos ( 4 x ) cos ( x )}{cos ( 4 x )}

Cancel the common factor:

cos (4 x)

= 2 cos (x)

= 2 cos (x)

2 cos (x) = 0

Divide both sides by

2

2 cos (x) = 0

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{0}{2}

Simplify

cos (x) = 0

cos (x) = 0

General solutions for

cos (x) = 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}

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

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

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{2}, x = \frac{3 π}{2}

Find the undefined points:

x = \frac{π}{8}, x = \frac{3 π}{8}, x = \frac{5 π}{8}, x = \frac{7 π}{8}, x = \frac{9 π}{8}, x = \frac{11 π}{8}, x = \frac{13 π}{8}, x = \frac{15 π}{8}

Find the zeros of the denominator

cos (4 x) = 0

General solutions for

cos (4 x) = 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}

4 x = \frac{π}{2} + 2 πn, 4 x = \frac{3 π}{2} + 2 πn

4 x = \frac{π}{2} + 2 πn, 4 x = \frac{3 π}{2} + 2 πn

Solve

4 x = \frac{π}{2} + 2 πn : x = \frac{π}{8} + \frac{πn}{2}

4 x = \frac{π}{2} + 2 πn

Divide both sides by

4

4 x = \frac{π}{2} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{π}{2}}{4} + \frac{2 πn}{4} : \frac{π}{8} + \frac{πn}{2}

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

\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{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}

Solve

4 x = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{8} + \frac{πn}{2}

4 x = \frac{3 π}{2} + 2 πn

Divide both sides by

4

4 x = \frac{3 π}{2} + 2 πn

Divide both sides by

4

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

Simplify

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

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{3 π}{2}}{4} + \frac{2 πn}{4} : \frac{3 π}{8} + \frac{πn}{2}

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

\frac{\frac{3 π}{2}}{4} = \frac{3 π}{8}

\frac{\frac{3 π}{2}}{4}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{3 π}{8}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{πn}{2}

x = \frac{π}{8} + \frac{πn}{2}, x = \frac{3 π}{8} + \frac{πn}{2}

Solutions for the range

0 \leq x < 2 π

x = \frac{π}{8}, x = \frac{3 π}{8}, x = \frac{5 π}{8}, x = \frac{7 π}{8}, x = \frac{9 π}{8}, x = \frac{11 π}{8}, x = \frac{13 π}{8}, x = \frac{15 π}{8}

\frac{π}{8}, \frac{3 π}{8}, \frac{π}{2}, \frac{5 π}{8}, \frac{7 π}{8}, \frac{9 π}{8}, \frac{11 π}{8}, \frac{3 π}{2}, \frac{13 π}{8}, \frac{15 π}{8}

Identify the intervals

0 < x < \frac{π}{8}, \frac{π}{8} < x < \frac{3 π}{8}, \frac{3 π}{8} < x < \frac{π}{2}, \frac{π}{2} < x < \frac{5 π}{8}, \frac{5 π}{8} < x < \frac{7 π}{8}, \frac{7 π}{8} < x < \frac{9 π}{8}, \frac{9 π}{8} < x < \frac{11 π}{8}, \frac{11 π}{8} < x < \frac{3 π}{2}, \frac{3 π}{2} < x < \frac{13 π}{8}, \frac{13 π}{8} < x < \frac{15 π}{8}, \frac{15 π}{8} < x < 2 π

Summarize in a table:

cos (5 x) + cos (3 x) cos (4 x) \frac{c o s ( 5 x ) + c o s ( 3 x )}{c o s ( 4 x )} x = 0 + + + 0 < x < \frac{π}{8} + + + x = \frac{π}{8} 00 Undefined \frac{π}{8} < x < \frac{3 π}{8} - - + x = \frac{3 π}{8} 00 Undefined \frac{3 π}{8} < x < \frac{π}{2} + + + x = \frac{π}{2} 0 + 0 \frac{π}{2} < x < \frac{5 π}{8} - + - x = \frac{5 π}{8} 00 Undefined \frac{5 π}{8} < x < \frac{7 π}{8} + - - x = \frac{7 π}{8} 00 Undefined \frac{7 π}{8} < x < \frac{9 π}{8} - + - x = \frac{9 π}{8} 00 Undefined \frac{9 π}{8} < x < \frac{11 π}{8} + - - x = \frac{11 π}{8} 00 Undefined \frac{11 π}{8} < x < \frac{3 π}{2} - + - x = \frac{3 π}{2} 0 + 0 \frac{3 π}{2} < x < \frac{13 π}{8} + + + x = \frac{13 π}{8} 00 Undefined \frac{13 π}{8} < x < \frac{15 π}{8} - - + x = \frac{15 π}{8} 00 Undefined \frac{15 π}{8} < x < 2 π + + + x = 2 π + + +

Identify the intervals that satisfy the required condition:

\leq 0

x = \frac{π}{2} or \frac{π}{2} < x < \frac{5 π}{8} or \frac{5 π}{8} < x < \frac{7 π}{8} or \frac{7 π}{8} < x < \frac{9 π}{8} or \frac{9 π}{8} < x < \frac{11 π}{8} or \frac{11 π}{8} < x < \frac{3 π}{2} or x = \frac{3 π}{2}

Merge Overlapping Intervals

\frac{π}{2} \leq x < \frac{5 π}{8} or \frac{5 π}{8} < x < \frac{7 π}{8} or \frac{7 π}{8} < x < \frac{9 π}{8} or \frac{9 π}{8} < x < \frac{11 π}{8} or \frac{11 π}{8} < x < \frac{3 π}{2} or x = \frac{3 π}{2}