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Popular >

4+4sin(x)-cos^2(x)>= 0

Solution

4 + 4 sin (x) - cos^{2} (x) \geq 0

Solution

True for all x \in R

Interval Notation

(- \infty, \infty)

Solution steps

4 + 4 sin (x) - cos^{2} (x) \geq 0

Use the following identity:

cos^{2} (x) + sin^{2} (x) = 1

Therefore

cos^{2} (x) = 1 - sin^{2} (x)

4 + 4 sin (x) - (1 - sin^{2} (x)) \geq 0

Simplify

sin^{2} (x) + 4 sin (x) + 3 \geq 0

Let:

u = sin (x)

u^{2} + 4 u + 3 \geq 0

u^{2} + 4 u + 3 \geq 0 : u \leq - 3 or u \geq - 1

u^{2} + 4 u + 3 \geq 0

Factor

u^{2} + 4 u + 3 : (u + 1) (u + 3)

u^{2} + 4 u + 3

Break the expression into groups

u^{2} + 4 u + 3

Definition

Factors of

3 : 1, 3

3

Divisors (Factors)

Find the Prime factors of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Add 1

1

The factors of

3

1, 3

For every two factors such that

u * v = 3,

check if

u + v = 4

Check

u = 1, v = 3 : u * v = 3, u + v = 4 \Rightarrow

True

u = 1, v = 3

Group into

(a x^{2} + ux) + (vx + c)

(u^{2} + u) + (3 u + 3)

= (u^{2} + u) + (3 u + 3)

Factor out

u

from

u^{2} + u : u (u + 1)

u^{2} + u

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{2} = uu

= uu + u

Factor out common term

u

= u (u + 1)

Factor out

3

from

3 u + 3 : 3 (u + 1)

3 u + 3

Factor out common term

3

= 3 (u + 1)

= u (u + 1) + 3 (u + 1)

Factor out common term

u + 1

= (u + 1) (u + 3)

(u + 1) (u + 3) \geq 0

Identify the intervals

Find the signs of the factors of

(u + 1) (u + 3)

Find the signs of

u + 1

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

u + 1 < 0 : u < - 1

u + 1 < 0

Move

1

to the right side

u + 1 < 0

Subtract

1

from both sides

u + 1 - 1 < 0 - 1

Simplify

u < - 1

u < - 1

u + 1 > 0 : u > - 1

u + 1 > 0

Move

1

to the right side

u + 1 > 0

Subtract

1

from both sides

u + 1 - 1 > 0 - 1

Simplify

u > - 1

u > - 1

Find the signs of

u + 3

u + 3 = 0 : u = - 3

u + 3 = 0

Move

3

to the right side

u + 3 = 0

Subtract

3

from both sides

u + 3 - 3 = 0 - 3

Simplify

u = - 3

u = - 3

u + 3 < 0 : u < - 3

u + 3 < 0

Move

3

to the right side

u + 3 < 0

Subtract

3

from both sides

u + 3 - 3 < 0 - 3

Simplify

u < - 3

u < - 3

u + 3 > 0 : u > - 3

u + 3 > 0

Move

3

to the right side

u + 3 > 0

Subtract

3

from both sides

u + 3 - 3 > 0 - 3

Simplify

u > - 3

u > - 3

Summarize in a table:

u + 1 u + 3 (u + 1) (u + 3) u < - 3 - - + u = - 3 - 00 - 3 < u < - 1 - + - u = - 1 0 + 0 u > - 1 + + +

Identify the intervals that satisfy the required condition:

\geq 0

u < - 3 or u = - 3 or u = - 1 or u > - 1

Merge Overlapping Intervals

u \leq - 3 or u = - 1 or u > - 1

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

u < - 3

u = - 3

u \leq - 3

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

u \leq - 3

u = - 1

u \leq - 3 or u = - 1

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

u \leq - 3 or u = - 1

u > - 1

u \leq - 3 or u \geq - 1

u \leq - 3 or u \geq - 1

u \leq - 3 or u \geq - 1

u \leq - 3 or u \geq - 1

Substitute back

u = sin (x)

sin (x) \leq - 3 or sin (x) \geq - 1

sin (x) \leq - 3 :

False for all

x \in R

sin (x) \leq - 3

Range of

sin (x) : - 1 \leq sin (x) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (x) \leq 1

- 1 \leq sin (x) \leq 1

sin (x) \leq - 3 and - 1 \leq sin (x) \leq 1 :

False

Let

y = sin (x)

Combine the intervals

y \leq - 3 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \leq - 3 and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y \leq - 3

and

- 1 \leq y \leq 1

False for all y \in R

False for all y \in R

No Solution for x \in R

False for all x \in R

sin (x) \geq - 1 :

True for all

x \in R

sin (x) \geq - 1

Range of

sin (x) : - 1 \leq sin (x) \leq 1

Function range definition

The range of the basic

sin

function is

- 1 \leq sin (x) \leq 1

- 1 \leq sin (x) \leq 1

sin (x) \geq - 1 and - 1 \leq sin (x) \leq 1 : - 1 \leq sin (x) \leq 1

Let

y = sin (x)

Combine the intervals

y \geq - 1 and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \geq - 1 and - 1 \leq y \leq 1

The intersection of two intervals is the set of numbers which are in both intervals

y \geq - 1

and

- 1 \leq y \leq 1

- 1 \leq y \leq 1

- 1 \leq y \leq 1

True for all x

True for all x \in R

Combine the intervals

False for all x \in R or True for all x \in R

Merge Overlapping Intervals

False for all x \in R or True for all x \in R

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

False for all

x \in R

True for all

x \in R

True for all x \in R

True for all x

True for all x \in R

Popular Examples

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2 cos^{2} (x) + cos (x) \leq 0

sin(2t)>=-(sqrt(3))/2

sin (2 t) \geq - \frac{3}{2}

2cos^2(x)+cos(x)<0

2 cos^{2} (x) + cos (x) < 0

3sin(x)<= 3/2

3 sin (x) \leq \frac{3}{2}

tan(x)>-1

tan (x) > - 1