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Popular >

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

Solution

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

Solution

2 πn \leq x \leq 2 π + 2 πn

Interval Notation

[2 πn, 2 π + 2 πn]

Decimal Notation

2 πn \leq x \leq 6.28318 \dots + 2 πn

Solution steps

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

Periodicity of

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

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

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

for

0 \leq x < 2 π

To find the zeroes, set the inequality to zero

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

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

No Solution for

x \in R

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

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

cos (x) - 2 = 0

Move

2

to the right side

cos (x) - 2 = 0

Add

2

to both sides

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

Simplify

cos (x) = 2

cos (x) = 2

- 1 \leq cos (x) \leq 1

No Solution for x \in R

Find the undefined points:

No Solution

Find the zeros of the denominator

sin (x) - 3 = 0

Move

3

to the right side

sin (x) - 3 = 0

Add

3

to both sides

sin (x) - 3 + 3 = 0 + 3

Simplify

sin (x) = 3

sin (x) = 3

- 1 \leq sin (x) \leq 1

No Solution for x \in R

Identify the intervals

0 < x < 2 π

Summarize in a table:

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

Identify the intervals that satisfy the required condition:

\geq 0

x = 0 or 0 < x < 2 π or x = 2 π

Merge Overlapping Intervals

0 \leq x < 2 π or x = 2 π

The union of two intervals is the set of numbers which are in either interval

x = 0

0 < x < 2 π

0 \leq x < 2 π

The union of two intervals is the set of numbers which are in either interval

0 \leq x < 2 π

x = 2 π

0 \leq x \leq 2 π

0 \leq x \leq 2 π

Apply the periodicity of

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

2 πn \leq x \leq 2 π + 2 πn

Popular Examples

cot(x)>0

cot (x) > 0

(3tan(x)-tan^3(x))/(1-tan^2(x))>0

\frac{3 tan ( x ) - tan ^{3} ( x )}{1 - tan ^{2} ( x )} > 0

cos(x)+1/2 >0,0pi<= x<= 2pi

cos (x) + \frac{1}{2} > 0, 0 π \leq x \leq 2 π

cos(x)>32

cos (x) > 32

0.5sin(x+pi/6)>= (sqrt(3))/4

0.5 sin (x + \frac{π}{6}) \geq \frac{3}{4}