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Popular >

(2*cos(x)-3)/(sin(x))>= 0

Solution

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

Solution

π + 2 πn < x < 2 π + 2 πn

Interval Notation

(π + 2 πn, 2 π + 2 πn)

Decimal Notation

3.14159 \dots + 2 πn < x < 6.28318 \dots + 2 πn

Solution steps

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

Periodicity of

\frac{2 cos ( x ) - 3}{sin ( x )} : 2 π

\frac{2 cos ( x ) - 3}{sin ( x )}

is composed of the following functions and periods:

cos (x)

with periodicity of

2 π

The compound periodicity is:

= 2 π

Find the zeroes and undifined points of

\frac{2 cos ( x ) - 3}{sin ( x )}

for

0 \leq x < 2 π

To find the zeroes, set the inequality to zero

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

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

No Solution for

x \in R

\frac{2 cos ( x ) - 3}{sin ( x )} = 0, 0 \leq x < 2 π

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

2 cos (x) - 3 = 0

Move

3

to the right side

2 cos (x) - 3 = 0

Add

3

to both sides

2 cos (x) - 3 + 3 = 0 + 3

Simplify

2 cos (x) = 3

2 cos (x) = 3

Divide both sides by

2

2 cos (x) = 3

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{3}{2}

Simplify

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2}

- 1 \leq cos (x) \leq 1

No Solution for x \in R

Find the undefined points:

x = 0, x = π

Find the zeros of the denominator

sin (x) = 0

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

0, π

Identify the intervals

0 < x < π, π < x < 2 π

Summarize in a table:

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

Identify the intervals that satisfy the required condition:

\geq 0

π < x < 2 π

Apply the periodicity of

\frac{2 cos ( x ) - 3}{sin ( x )}

π + 2 πn < x < 2 π + 2 πn

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