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Popular >

(sin(x)(2cos(x)+1))/(sin(x)+cos(x))<0

Solution

\frac{sin ( x ) ( 2 cos ( x ) + 1 )}{sin ( x ) + cos ( x )} < 0

Solution

\frac{2 π}{3} + 2 πn < x < \frac{3 π}{4} + 2 πn or π + 2 πn < x < \frac{4 π}{3} + 2 πn or \frac{7 π}{4} + 2 πn < x < 2 π + 2 πn

Interval Notation

(\frac{2 π}{3} + 2 πn, \frac{3 π}{4} + 2 πn) \cup (π + 2 πn, \frac{4 π}{3} + 2 πn) \cup (\frac{7 π}{4} + 2 πn, 2 π + 2 πn)

Decimal Notation

2.09439 \dots + 2 πn < x < 2.35619 \dots + 2 πn or 3.14159 \dots + 2 πn < x < 4.18879 \dots + 2 πn or 5.49778 \dots + 2 πn < x < 6.28318 \dots + 2 πn

Solution steps

\frac{sin ( x ) ( 2 cos ( x ) + 1 )}{sin ( x ) + cos ( x )} < 0

Periodicity of

\frac{sin ( x ) ( 2 cos ( x ) + 1 )}{sin ( x ) + cos ( x )} : 2 π

\frac{sin ( x ) ( 2 cos ( x ) + 1 )}{sin ( x ) + cos ( x )}

is composed of the following functions and periods:

sin (x)

with periodicity of

2 π

The compound periodicity is:

= 2 π

Find the zeroes and undifined points of

\frac{sin ( x ) ( 2 cos ( x ) + 1 )}{sin ( x ) + cos ( x )}

for

0 \leq x < 2 π

To find the zeroes, set the inequality to zero

\frac{sin ( x ) ( 2 cos ( x ) + 1 )}{sin ( x ) + cos ( x )} = 0

\frac{sin ( x ) ( 2 cos ( x ) + 1 )}{sin ( x ) + cos ( x )} = 0, 0 \leq x < 2 π : x = 0, x = π, x = \frac{2 π}{3}, x = \frac{4 π}{3}

\frac{sin ( x ) ( 2 cos ( x ) + 1 )}{sin ( x ) + cos ( x )} = 0, 0 \leq x < 2 π

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin (x) (2 cos (x) + 1) = 0

Solving each part separately

sin (x) = 0 or 2 cos (x) + 1 = 0

sin (x) = 0, 0 \leq x < 2 π : x = 0, x = π

sin (x) = 0, 0 \leq x < 2 π

General solutions for

sin (x) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

Solutions for the range

0 \leq x < 2 π

x = 0, x = π

2 cos (x) + 1 = 0, 0 \leq x < 2 π : x = \frac{2 π}{3}, x = \frac{4 π}{3}

2 cos (x) + 1 = 0, 0 \leq x < 2 π

Move

1

to the right side

2 cos (x) + 1 = 0

Subtract

1

from both sides

2 cos (x) + 1 - 1 = 0 - 1

Simplify

2 cos (x) = - 1

2 cos (x) = - 1

Divide both sides by

2

2 cos (x) = - 1

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{- 1}{2}

Simplify

cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{2}

General solutions for

cos (x) = - \frac{1}{2}

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

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

Solutions for the range

0 \leq x < 2 π

x = \frac{2 π}{3}, x = \frac{4 π}{3}

Combine all the solutions

x = 0, x = π, x = \frac{2 π}{3}, x = \frac{4 π}{3}

Find the undefined points:

x = \frac{3 π}{4}, x = \frac{7 π}{4}

Find the zeros of the denominator

sin (x) + cos (x) = 0

Rewrite using trig identities

sin (x) + cos (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{sin ( x ) + cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{sin ( x )}{cos ( x )} + 1 = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

tan (x) + 1 = 0

tan (x) + 1 = 0

Move

1

to the right side

tan (x) + 1 = 0

Subtract

1

from both sides

tan (x) + 1 - 1 = 0 - 1

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Solutions for the range

0 \leq x < 2 π

x = \frac{3 π}{4}, x = \frac{7 π}{4}

0, \frac{2 π}{3}, \frac{3 π}{4}, π, \frac{4 π}{3}, \frac{7 π}{4}

Identify the intervals

0 < x < \frac{2 π}{3}, \frac{2 π}{3} < x < \frac{3 π}{4}, \frac{3 π}{4} < x < π, π < x < \frac{4 π}{3}, \frac{4 π}{3} < x < \frac{7 π}{4}, \frac{7 π}{4} < x < 2 π

Summarize in a table:

sin (x) 2 cos (x) + 1 sin (x) + cos (x) \frac{s i n ( x ) ( 2 c o s ( x ) + 1 )}{s i n ( x ) + c o s ( x )} x = 0 0 + + 0 0 < x < \frac{2 π}{3} + + + + x = \frac{2 π}{3} + 0 + 0 \frac{2 π}{3} < x < \frac{3 π}{4} + - + - x = \frac{3 π}{4} + - 0 Undefined \frac{3 π}{4} < x < π + - - + x = π 0 - - 0 π < x < \frac{4 π}{3} - - - - x = \frac{4 π}{3} - 0 - 0 \frac{4 π}{3} < x < \frac{7 π}{4} - + - + x = \frac{7 π}{4} - + 0 Undefined \frac{7 π}{4} < x < 2 π - + + - x = 2 π 0 + + 0

Identify the intervals that satisfy the required condition:

< 0

\frac{2 π}{3} < x < \frac{3 π}{4} or π < x < \frac{4 π}{3} or \frac{7 π}{4} < x < 2 π

Apply the periodicity of

\frac{sin ( x ) ( 2 cos ( x ) + 1 )}{sin ( x ) + cos ( x )}

\frac{2 π}{3} + 2 πn < x < \frac{3 π}{4} + 2 πn or π + 2 πn < x < \frac{4 π}{3} + 2 πn or \frac{7 π}{4} + 2 πn < x < 2 π + 2 πn

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