English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cos^2(x)-sin^2(x)+sqrt(3)cos(x)-2<= 0

Solution

cos^{2} (x) - sin^{2} (x) + 3 cos (x) - 2 \leq 0

Solution

\frac{π}{6} + 2 πn \leq x \leq \frac{11 π}{6} + 2 πn

Interval Notation

[\frac{π}{6} + 2 πn, \frac{11 π}{6} + 2 πn]

Decimal Notation

0.52359 \dots + 2 πn \leq x \leq 5.75958 \dots + 2 πn

Solution steps

cos^{2} (x) - sin^{2} (x) + 3 cos (x) - 2 \leq 0

Use the following identity:

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

Therefore

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

cos^{2} (x) - (1 - cos^{2} (x)) + 3 cos (x) - 2 \leq 0

Simplify

cos^{2} (x) - (1 - cos^{2} (x)) + 3 cos (x) - 2 : 2 cos^{2} (x) + 3 cos (x) - 3

cos^{2} (x) - (1 - cos^{2} (x)) + 3 cos (x) - 2

- (1 - cos^{2} (x)) : - 1 + cos^{2} (x)

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

= cos^{2} (x) - 1 + cos^{2} (x) + 3 cos (x) - 2

Simplify

cos^{2} (x) - 1 + cos^{2} (x) + 3 cos (x) - 2 : 2 cos^{2} (x) + 3 cos (x) - 3

cos^{2} (x) - 1 + cos^{2} (x) + 3 cos (x) - 2

Group like terms

= cos^{2} (x) + cos^{2} (x) + 3 cos (x) - 1 - 2

Add similar elements:

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

= 2 cos^{2} (x) + 3 cos (x) - 1 - 2

Subtract the numbers:

- 1 - 2 = - 3

= 2 cos^{2} (x) + 3 cos (x) - 3

= 2 cos^{2} (x) + 3 cos (x) - 3

2 cos^{2} (x) + 3 cos (x) - 3 \leq 0

Let:

u = cos (x)

2 u^{2} + 3 u - 3 \leq 0

2 u^{2} + 3 u - 3 \leq 0 : - 3 \leq u \leq \frac{3}{2}

2 u^{2} + 3 u - 3 \leq 0

Complete the square

2 u^{2} + 3 u - 3 : 2 (u + \frac{3}{4})^{2} - \frac{27}{8}

2 u^{2} + 3 u - 3

Write

2 u^{2} + 3 u - 3

in the form:

x^{2} + 2 a x + a^{2}

Factor out

2

2 (u^{2} + \frac{3 u}{2} - \frac{3}{2})

2 a = \frac{3}{2} : a = \frac{3}{4}

2 a = \frac{3}{2}

Divide both sides by

2

2 a = \frac{3}{2}

Divide both sides by

2

\frac{2 a}{2} = \frac{\frac{3}{2}}{2}

Simplify

\frac{2 a}{2} = \frac{\frac{3}{2}}{2}

Simplify

\frac{2 a}{2} : a

\frac{2 a}{2}

Divide the numbers:

\frac{2}{2} = 1

= a

Simplify

\frac{\frac{3}{2}}{2} : \frac{3}{4}

\frac{\frac{3}{2}}{2}

Apply the fraction rule:

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

= \frac{3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

a = \frac{3}{4}

a = \frac{3}{4}

a = \frac{3}{4}

Add and subtract

(\frac{3}{4})^{2}

2 u^{2} + \frac{3 u}{2} - \frac{3}{2} + (\frac{3}{4})^{2} - (\frac{3}{4})^{2}

x^{2} + 2 a x + a^{2} = (x + a)^{2}

u^{2} + \frac{3}{2} u + (\frac{3}{4})^{2} = (u + \frac{3}{4})^{2}

2 (u + \frac{3}{4})^{2} - \frac{3}{2} - (\frac{3}{4})^{2}

Simplify

2 (u + \frac{3}{4})^{2} - \frac{27}{8}

2 (u + \frac{3}{4})^{2} - \frac{27}{8} \leq 0

Factor

2 u^{2} + 3 u - 3 : (u - \frac{3}{2}) (u + 3)

2 u^{2} + 3 u - 3

A quadratic of the form:

a x^{2} + b x + c

with roots

x_{1}, x_{2},

can be written as

(x - x_{1}) (x - x_{2})

2 u^{2} + 3 u - 3 = 0 : u = \frac{3}{2}, u = - 3

2 u^{2} + 3 u - 3 = 0

Solve with the quadratic formula

2 u^{2} + 3 u - 3 = 0

Quadratic Equation Formula:

For

a = 2, b = 3, c = - 3

u_{1, 2} = \frac{- 3 \pm ( 3 ) ^{2} - 4 \cdot 2 ( - 3 )}{2 \cdot 2}

u_{1, 2} = \frac{- 3 \pm ( 3 ) ^{2} - 4 \cdot 2 ( - 3 )}{2 \cdot 2}

(3)^{2} - 4 \cdot 2 (- 3) = 33

(3)^{2} - 4 \cdot 2 (- 3)

Apply rule

- (- a) = a

= (3)^{2} + 4 \cdot 2 \cdot 3

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

4 \cdot 2 \cdot 3 = 24

4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 24

= 3 + 24

Add the numbers:

3 + 24 = 27

= 27

Prime factorization of

27 : 3^{3}

27

27

divides by

327 = 9 \cdot 3

= 3 \cdot 9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3 \cdot 3

3

is a prime number, therefore no further factorization is possible

= 3 \cdot 3 \cdot 3

= 3^{3}

= 3^{3}

Apply exponent rule:

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

= 3^{2} \cdot 3

Apply radical rule:

= 3 3^{2}

Apply radical rule:

3^{2} = 3

= 33

u_{1, 2} = \frac{- 3 \pm 3 3}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 3 + 3 3}{2 \cdot 2}, u_{2} = \frac{- 3 - 3 3}{2 \cdot 2}

u = \frac{- 3 + 3 3}{2 \cdot 2} : \frac{3}{2}

\frac{- 3 + 3 3}{2 \cdot 2}

Add similar elements:

- 3 + 33 = 23

= \frac{2 3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 3}{4}

Cancel the common factor:

2

= \frac{3}{2}

u = \frac{- 3 - 3 3}{2 \cdot 2} : - 3

\frac{- 3 - 3 3}{2 \cdot 2}

Add similar elements:

- 3 - 33 = - 43

= \frac{- 4 3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 4 3}{4}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4 3}{4}

Divide the numbers:

\frac{4}{4} = 1

= - 3

The solutions to the quadratic equation are:

u = \frac{3}{2}, u = - 3

2 u^{2} + 3 u - 3 = (u - \frac{3}{2}) (u + 3)

2 (u + \frac{3}{4})^{2} - \frac{27}{8} \leq 0

Identify the intervals

Find the signs of the factors of

2 (u + \frac{3}{4})^{2} - \frac{27}{8}

Find the signs of

2 (u + \frac{3}{4})^{2} - \frac{27}{8}

2 (u + \frac{3}{4})^{2} - \frac{27}{8} = 0 : u = - 3 or u = \frac{3}{2}

2 (u + \frac{3}{4})^{2} - \frac{27}{8} = 0

Move

\frac{27}{8}

to the right side

2 (u + \frac{3}{4})^{2} - \frac{27}{8} = 0

Add

\frac{27}{8}

to both sides

2 (u + \frac{3}{4})^{2} - \frac{27}{8} + \frac{27}{8} = 0 + \frac{27}{8}

Simplify

2 (u + \frac{3}{4})^{2} = \frac{27}{8}

2 (u + \frac{3}{4})^{2} = \frac{27}{8}

Divide both sides by

2

2 (u + \frac{3}{4})^{2} = \frac{27}{8}

Divide both sides by

2

\frac{2 ( u + \frac{3}{4} ) ^{2}}{2} = \frac{\frac{27}{8}}{2}

Simplify

\frac{2 ( u + \frac{3}{4} ) ^{2}}{2} = \frac{\frac{27}{8}}{2}

Simplify

\frac{2 ( u + \frac{3}{4} ) ^{2}}{2} : (u + \frac{3}{4})^{2}

\frac{2 ( u + \frac{3}{4} ) ^{2}}{2}

Divide the numbers:

\frac{2}{2} = 1

= (u + \frac{3}{4})^{2}

Simplify

\frac{\frac{27}{8}}{2} : \frac{27}{16}

\frac{\frac{27}{8}}{2}

Apply the fraction rule:

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

= \frac{27}{8 \cdot 2}

Multiply the numbers:

8 \cdot 2 = 16

= \frac{27}{16}

(u + \frac{3}{4})^{2} = \frac{27}{16}

(u + \frac{3}{4})^{2} = \frac{27}{16}

(u + \frac{3}{4})^{2} = \frac{27}{16}

For

(g (x))^{2} = f (a)

the solutions are

g (x) = f (a), - f (a)

Solve

u + \frac{3}{4} = \frac{27}{16} : u = \frac{3}{2}

u + \frac{3}{4} = \frac{27}{16}

Simplify

\frac{27}{16} : \frac{3 3}{4}

\frac{27}{16}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{27}{16}

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

= \frac{27}{4}

27 = 33

27

Prime factorization of

27 : 3^{3}

27

27

divides by

327 = 9 \cdot 3

= 3 \cdot 9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3 \cdot 3

3

is a prime number, therefore no further factorization is possible

= 3 \cdot 3 \cdot 3

= 3^{3}

= 3^{3}

Apply exponent rule:

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

= 3^{2} \cdot 3

Apply radical rule:

= 3 3^{2}

Apply radical rule:

3^{2} = 3

= 33

= \frac{3 3}{4}

u + \frac{3}{4} = \frac{3 3}{4}

Move

\frac{3}{4}

to the right side

u + \frac{3}{4} = \frac{3 3}{4}

Subtract

\frac{3}{4}

from both sides

u + \frac{3}{4} - \frac{3}{4} = \frac{3 3}{4} - \frac{3}{4}

Simplify

u + \frac{3}{4} - \frac{3}{4} = \frac{3 3}{4} - \frac{3}{4}

Simplify

u + \frac{3}{4} - \frac{3}{4} : u

u + \frac{3}{4} - \frac{3}{4}

Add similar elements:

\frac{3}{4} - \frac{3}{4} = 0

= u

Simplify

\frac{3 3}{4} - \frac{3}{4} : \frac{3}{2}

\frac{3 3}{4} - \frac{3}{4}

Apply rule

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

= \frac{3 3 - 3}{4}

Add similar elements:

33 - 3 = 23

= \frac{2 3}{4}

Cancel the common factor:

2

= \frac{3}{2}

u = \frac{3}{2}

u = \frac{3}{2}

u = \frac{3}{2}

Solve

u + \frac{3}{4} = - \frac{27}{16} : u = - 3

u + \frac{3}{4} = - \frac{27}{16}

Simplify

- \frac{27}{16} : - \frac{3 3}{4}

- \frac{27}{16}

Simplify

\frac{27}{16} : \frac{3 3}{4}

\frac{27}{16}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{27}{16}

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

= \frac{27}{4}

27 = 33

27

Prime factorization of

27 : 3^{3}

27

27

divides by

327 = 9 \cdot 3

= 3 \cdot 9

9

divides by

39 = 3 \cdot 3

= 3 \cdot 3 \cdot 3

3

is a prime number, therefore no further factorization is possible

= 3 \cdot 3 \cdot 3

= 3^{3}

= 3^{3}

Apply exponent rule:

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

= 3^{2} \cdot 3

Apply radical rule:

= 3 3^{2}

Apply radical rule:

3^{2} = 3

= 33

= \frac{3 3}{4}

= - \frac{3 3}{4}

u + \frac{3}{4} = - \frac{3 3}{4}

Move

\frac{3}{4}

to the right side

u + \frac{3}{4} = - \frac{3 3}{4}

Subtract

\frac{3}{4}

from both sides

u + \frac{3}{4} - \frac{3}{4} = - \frac{3 3}{4} - \frac{3}{4}

Simplify

u + \frac{3}{4} - \frac{3}{4} = - \frac{3 3}{4} - \frac{3}{4}

Simplify

u + \frac{3}{4} - \frac{3}{4} : u

u + \frac{3}{4} - \frac{3}{4}

Add similar elements:

\frac{3}{4} - \frac{3}{4} = 0

= u

Simplify

- \frac{3 3}{4} - \frac{3}{4} : - 3

- \frac{3 3}{4} - \frac{3}{4}

Apply rule

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

= \frac{- 3 3 - 3}{4}

Add similar elements:

- 33 - 3 = - 43

= \frac{- 4 3}{4}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4 3}{4}

Divide the numbers:

\frac{4}{4} = 1

= - 3

u = - 3

u = - 3

u = - 3

The solutions to the quadratic equation are:

u = \frac{3}{2}, u = - 3

2 (u + \frac{3}{4})^{2} - \frac{27}{8} < 0 : - 3 < u < \frac{3}{2}

2 (u + \frac{3}{4})^{2} - \frac{27}{8} < 0

Factor

2 (u + \frac{3}{4})^{2} - \frac{27}{8} : (2 u + 6) (2 u - \frac{3}{2})

2 (u + \frac{3}{4})^{2} - \frac{27}{8}

Rewrite

2 (u + \frac{3}{4})^{2} - \frac{27}{8}

(2 (u + \frac{3}{4}))^{2} - (\frac{27}{8})^{2}

2 (u + \frac{3}{4})^{2} - \frac{27}{8}

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} (u + \frac{3}{4})^{2} - \frac{27}{8}

Apply radical rule:

a = (a)^{2}

\frac{27}{8} = (\frac{27}{8})^{2}

= (2)^{2} (u + \frac{3}{4})^{2} - (\frac{27}{8})^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(2)^{2} (u + \frac{3}{4})^{2} = (2 (u + \frac{3}{4}))^{2}

= (2 (u + \frac{3}{4}))^{2} - (\frac{27}{8})^{2}

= (2 (u + \frac{3}{4}))^{2} - (\frac{27}{8})^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(2 (u + \frac{3}{4}))^{2} - (\frac{27}{8})^{2} = (2 (u + \frac{3}{4}) + \frac{27}{8}) (2 (u + \frac{3}{4}) - \frac{27}{8})

= (2 (u + \frac{3}{4}) + \frac{27}{8}) (2 (u + \frac{3}{4}) - \frac{27}{8})

Refine

= (2 (u + \frac{3}{4}) + \frac{3 3}{2 2}) (2 (u + \frac{3}{4}) - \frac{3 3}{2 2})

Expand

2 (u + \frac{3}{4}) + \frac{3 3}{2 2} : 2 u + 6

2 (u + \frac{3}{4}) + \frac{3 3}{2 2}

Expand

2 (u + \frac{3}{4}) : 2 u + \frac{3}{2 2}

2 (u + \frac{3}{4})

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = u, c = \frac{3}{4}

= 2 u + 2 \frac{3}{4}

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

2 \frac{3}{4}

Multiply fractions:

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

= \frac{3 2}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{3 2}{2 ^{2}}

Cancel

\frac{3 2}{2 ^{2}} : \frac{3}{2 ^{\frac{3}{2}}}

\frac{3 2}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{3 \cdot 2 ^{\frac{1}{2}}}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{3}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{3}{2 ^{\frac{3}{2}}}

= \frac{3}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= \frac{3}{2 2}

= 2 u + \frac{3}{2 2}

= 2 u + \frac{3}{2 2} + \frac{3 3}{2 2}

Combine the fractions

\frac{3}{2 2} + \frac{3 3}{2 2} : 6

Apply rule

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

= \frac{3 + 3 3}{2 2}

Add similar elements:

3 + 33 = 43

= \frac{4 3}{2 2}

Divide the numbers:

\frac{4}{2} = 2

= \frac{2 3}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 3}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 3 \cdot 2^{- \frac{1}{2} + 1}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}} 3

Apply radical rule:

2^{\frac{1}{2}} = 2

= 23

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= 2 u + 6

= (2 u + 6) (2 (u + \frac{3}{4}) - \frac{3 3}{2 2})

Expand

2 (u + \frac{3}{4}) - \frac{3 3}{2 2} : 2 u - \frac{3}{2}

2 (u + \frac{3}{4}) - \frac{3 3}{2 2}

Expand

2 (u + \frac{3}{4}) : 2 u + \frac{3}{2 2}

2 (u + \frac{3}{4})

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = u, c = \frac{3}{4}

= 2 u + 2 \frac{3}{4}

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

2 \frac{3}{4}

Multiply fractions:

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

= \frac{3 2}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{3 2}{2 ^{2}}

Cancel

\frac{3 2}{2 ^{2}} : \frac{3}{2 ^{\frac{3}{2}}}

\frac{3 2}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{3 \cdot 2 ^{\frac{1}{2}}}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{3}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{3}{2 ^{\frac{3}{2}}}

= \frac{3}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= \frac{3}{2 2}

= 2 u + \frac{3}{2 2}

= 2 u + \frac{3}{2 2} - \frac{3 3}{2 2}

Combine the fractions

\frac{3}{2 2} - \frac{3 3}{2 2} : - \frac{3}{2}

Apply rule

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

= \frac{3 - 3 3}{2 2}

Add similar elements:

3 - 33 = - 23

= \frac{- 2 3}{2 2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 3}{2 2}

Divide the numbers:

\frac{2}{2} = 1

= - \frac{3}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{3}{2}

= 2 u - \frac{3}{2}

= (2 u + 6) (2 u - \frac{3}{2})

(2 u + 6) (2 u - \frac{3}{2}) < 0

Identify the intervals

Find the signs of the factors of

(2 u + 6) (2 u - \frac{3}{2})

Find the signs of

2 u + 6

2 u + 6 = 0 : u = - 3

2 u + 6 = 0

Move

6

to the right side

2 u + 6 = 0

Subtract

6

from both sides

2 u + 6 - 6 = 0 - 6

Simplify

2 u = - 6

2 u = - 6

Divide both sides by

2

2 u = - 6

Divide both sides by

2

\frac{2 u}{2} = \frac{- 6}{2}

Simplify

\frac{2 u}{2} = \frac{- 6}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 6}{2} : - 3

\frac{- 6}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{2}

Cancel

\frac{6}{2} : 3

\frac{6}{2}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 3}{2}

Cancel the common factor:

2

= 3

= - 3

u = - 3

u = - 3

u = - 3

2 u + 6 < 0 : u < - 3

2 u + 6 < 0

Move

6

to the right side

2 u + 6 < 0

Subtract

6

from both sides

2 u + 6 - 6 < 0 - 6

Simplify

2 u < - 6

2 u < - 6

Divide both sides by

2

2 u < - 6

Divide both sides by

2

\frac{2 u}{2} < \frac{- 6}{2}

Simplify

\frac{2 u}{2} < \frac{- 6}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 6}{2} : - 3

\frac{- 6}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{2}

Cancel

\frac{6}{2} : 3

\frac{6}{2}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 3}{2}

Cancel the common factor:

2

= 3

= - 3

u < - 3

u < - 3

u < - 3

2 u + 6 > 0 : u > - 3

2 u + 6 > 0

Move

6

to the right side

2 u + 6 > 0

Subtract

6

from both sides

2 u + 6 - 6 > 0 - 6

Simplify

2 u > - 6

2 u > - 6

Divide both sides by

2

2 u > - 6

Divide both sides by

2

\frac{2 u}{2} > \frac{- 6}{2}

Simplify

\frac{2 u}{2} > \frac{- 6}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 6}{2} : - 3

\frac{- 6}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{2}

Cancel

\frac{6}{2} : 3

\frac{6}{2}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 3}{2}

Cancel the common factor:

2

= 3

= - 3

u > - 3

u > - 3

u > - 3

Find the signs of

2 u - \frac{3}{2}

2 u - \frac{3}{2} = 0 : u = \frac{3}{2}

2 u - \frac{3}{2} = 0

Move

\frac{3}{2}

to the right side

2 u - \frac{3}{2} = 0

Add

\frac{3}{2}

to both sides

2 u - \frac{3}{2} + \frac{3}{2} = 0 + \frac{3}{2}

Simplify

2 u = \frac{3}{2}

2 u = \frac{3}{2}

Divide both sides by

2

2 u = \frac{3}{2}

Divide both sides by

2

\frac{2 u}{2} = \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} = \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{\frac{3}{2}}{2} : \frac{3}{2}

\frac{\frac{3}{2}}{2}

\frac{3}{2} = \frac{3}{2}

\frac{3}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{2}

= \frac{\frac{3}{2}}{2}

Apply the fraction rule:

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

= \frac{3}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{3}{2}

u = \frac{3}{2}

u = \frac{3}{2}

u = \frac{3}{2}

2 u - \frac{3}{2} < 0 : u < \frac{3}{2}

2 u - \frac{3}{2} < 0

Move

\frac{3}{2}

to the right side

2 u - \frac{3}{2} < 0

Add

\frac{3}{2}

to both sides

2 u - \frac{3}{2} + \frac{3}{2} < 0 + \frac{3}{2}

Simplify

2 u < \frac{3}{2}

2 u < \frac{3}{2}

Divide both sides by

2

2 u < \frac{3}{2}

Divide both sides by

2

\frac{2 u}{2} < \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} < \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{\frac{3}{2}}{2} : \frac{3}{2}

\frac{\frac{3}{2}}{2}

\frac{3}{2} = \frac{3}{2}

\frac{3}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{2}

= \frac{\frac{3}{2}}{2}

Apply the fraction rule:

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

= \frac{3}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{3}{2}

u < \frac{3}{2}

u < \frac{3}{2}

u < \frac{3}{2}

2 u - \frac{3}{2} > 0 : u > \frac{3}{2}

2 u - \frac{3}{2} > 0

Move

\frac{3}{2}

to the right side

2 u - \frac{3}{2} > 0

Add

\frac{3}{2}

to both sides

2 u - \frac{3}{2} + \frac{3}{2} > 0 + \frac{3}{2}

Simplify

2 u > \frac{3}{2}

2 u > \frac{3}{2}

Divide both sides by

2

2 u > \frac{3}{2}

Divide both sides by

2

\frac{2 u}{2} > \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} > \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{\frac{3}{2}}{2} : \frac{3}{2}

\frac{\frac{3}{2}}{2}

\frac{3}{2} = \frac{3}{2}

\frac{3}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{2}

= \frac{\frac{3}{2}}{2}

Apply the fraction rule:

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

= \frac{3}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{3}{2}

u > \frac{3}{2}

u > \frac{3}{2}

u > \frac{3}{2}

Summarize in a table:

2 u + 6 2 u - \frac{3}{2} (2 u + 6) (2 u - \frac{3}{2}) u < - 3 - - + u = - 3 0 - 0 - 3 < u < \frac{3}{2} + - - u = \frac{3}{2} + 00 u > \frac{3}{2} + + +

Identify the intervals that satisfy the required condition:

< 0

- 3 < u < \frac{3}{2}

- 3 < u < \frac{3}{2}

2 (u + \frac{3}{4})^{2} - \frac{27}{8} > 0 : u < - 3 or u > \frac{3}{2}

2 (u + \frac{3}{4})^{2} - \frac{27}{8} > 0

Factor

2 (u + \frac{3}{4})^{2} - \frac{27}{8} : (2 u + 6) (2 u - \frac{3}{2})

2 (u + \frac{3}{4})^{2} - \frac{27}{8}

Rewrite

2 (u + \frac{3}{4})^{2} - \frac{27}{8}

(2 (u + \frac{3}{4}))^{2} - (\frac{27}{8})^{2}

2 (u + \frac{3}{4})^{2} - \frac{27}{8}

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} (u + \frac{3}{4})^{2} - \frac{27}{8}

Apply radical rule:

a = (a)^{2}

\frac{27}{8} = (\frac{27}{8})^{2}

= (2)^{2} (u + \frac{3}{4})^{2} - (\frac{27}{8})^{2}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(2)^{2} (u + \frac{3}{4})^{2} = (2 (u + \frac{3}{4}))^{2}

= (2 (u + \frac{3}{4}))^{2} - (\frac{27}{8})^{2}

= (2 (u + \frac{3}{4}))^{2} - (\frac{27}{8})^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(2 (u + \frac{3}{4}))^{2} - (\frac{27}{8})^{2} = (2 (u + \frac{3}{4}) + \frac{27}{8}) (2 (u + \frac{3}{4}) - \frac{27}{8})

= (2 (u + \frac{3}{4}) + \frac{27}{8}) (2 (u + \frac{3}{4}) - \frac{27}{8})

Refine

= (2 (u + \frac{3}{4}) + \frac{3 3}{2 2}) (2 (u + \frac{3}{4}) - \frac{3 3}{2 2})

Expand

2 (u + \frac{3}{4}) + \frac{3 3}{2 2} : 2 u + 6

2 (u + \frac{3}{4}) + \frac{3 3}{2 2}

Expand

2 (u + \frac{3}{4}) : 2 u + \frac{3}{2 2}

2 (u + \frac{3}{4})

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = u, c = \frac{3}{4}

= 2 u + 2 \frac{3}{4}

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

2 \frac{3}{4}

Multiply fractions:

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

= \frac{3 2}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{3 2}{2 ^{2}}

Cancel

\frac{3 2}{2 ^{2}} : \frac{3}{2 ^{\frac{3}{2}}}

\frac{3 2}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{3 \cdot 2 ^{\frac{1}{2}}}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{3}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{3}{2 ^{\frac{3}{2}}}

= \frac{3}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= \frac{3}{2 2}

= 2 u + \frac{3}{2 2}

= 2 u + \frac{3}{2 2} + \frac{3 3}{2 2}

Combine the fractions

\frac{3}{2 2} + \frac{3 3}{2 2} : 6

Apply rule

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

= \frac{3 + 3 3}{2 2}

Add similar elements:

3 + 33 = 43

= \frac{4 3}{2 2}

Divide the numbers:

\frac{4}{2} = 2

= \frac{2 3}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 3}{2 ^{\frac{1}{2}}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 3 \cdot 2^{- \frac{1}{2} + 1}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}} 3

Apply radical rule:

2^{\frac{1}{2}} = 2

= 23

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= 2 u + 6

= (2 u + 6) (2 (u + \frac{3}{4}) - \frac{3 3}{2 2})

Expand

2 (u + \frac{3}{4}) - \frac{3 3}{2 2} : 2 u - \frac{3}{2}

2 (u + \frac{3}{4}) - \frac{3 3}{2 2}

Expand

2 (u + \frac{3}{4}) : 2 u + \frac{3}{2 2}

2 (u + \frac{3}{4})

Apply the distributive law:

a (b + c) = ab + a c

a = 2, b = u, c = \frac{3}{4}

= 2 u + 2 \frac{3}{4}

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

2 \frac{3}{4}

Multiply fractions:

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

= \frac{3 2}{4}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

= \frac{3 2}{2 ^{2}}

Cancel

\frac{3 2}{2 ^{2}} : \frac{3}{2 ^{\frac{3}{2}}}

\frac{3 2}{2 ^{2}}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{3 \cdot 2 ^{\frac{1}{2}}}{2 ^{2}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{2}} = \frac{1}{2 ^{2 - \frac{1}{2}}}

= \frac{3}{2 ^{2 - \frac{1}{2}}}

Subtract the numbers:

2 - \frac{1}{2} = \frac{3}{2}

= \frac{3}{2 ^{\frac{3}{2}}}

= \frac{3}{2 ^{\frac{3}{2}}}

2^{\frac{3}{2}} = 22

2^{\frac{3}{2}}

2^{\frac{3}{2}} = 2^{1 + \frac{1}{2}}

= 2^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{1} \cdot 2^{\frac{1}{2}}

Refine

= 22

= \frac{3}{2 2}

= 2 u + \frac{3}{2 2}

= 2 u + \frac{3}{2 2} - \frac{3 3}{2 2}

Combine the fractions

\frac{3}{2 2} - \frac{3 3}{2 2} : - \frac{3}{2}

Apply rule

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

= \frac{3 - 3 3}{2 2}

Add similar elements:

3 - 33 = - 23

= \frac{- 2 3}{2 2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 3}{2 2}

Divide the numbers:

\frac{2}{2} = 1

= - \frac{3}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{3}{2}

= 2 u - \frac{3}{2}

= (2 u + 6) (2 u - \frac{3}{2})

(2 u + 6) (2 u - \frac{3}{2}) > 0

Identify the intervals

Find the signs of the factors of

(2 u + 6) (2 u - \frac{3}{2})

Find the signs of

2 u + 6

2 u + 6 = 0 : u = - 3

2 u + 6 = 0

Move

6

to the right side

2 u + 6 = 0

Subtract

6

from both sides

2 u + 6 - 6 = 0 - 6

Simplify

2 u = - 6

2 u = - 6

Divide both sides by

2

2 u = - 6

Divide both sides by

2

\frac{2 u}{2} = \frac{- 6}{2}

Simplify

\frac{2 u}{2} = \frac{- 6}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 6}{2} : - 3

\frac{- 6}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{2}

Cancel

\frac{6}{2} : 3

\frac{6}{2}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 3}{2}

Cancel the common factor:

2

= 3

= - 3

u = - 3

u = - 3

u = - 3

2 u + 6 < 0 : u < - 3

2 u + 6 < 0

Move

6

to the right side

2 u + 6 < 0

Subtract

6

from both sides

2 u + 6 - 6 < 0 - 6

Simplify

2 u < - 6

2 u < - 6

Divide both sides by

2

2 u < - 6

Divide both sides by

2

\frac{2 u}{2} < \frac{- 6}{2}

Simplify

\frac{2 u}{2} < \frac{- 6}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 6}{2} : - 3

\frac{- 6}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{2}

Cancel

\frac{6}{2} : 3

\frac{6}{2}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 3}{2}

Cancel the common factor:

2

= 3

= - 3

u < - 3

u < - 3

u < - 3

2 u + 6 > 0 : u > - 3

2 u + 6 > 0

Move

6

to the right side

2 u + 6 > 0

Subtract

6

from both sides

2 u + 6 - 6 > 0 - 6

Simplify

2 u > - 6

2 u > - 6

Divide both sides by

2

2 u > - 6

Divide both sides by

2

\frac{2 u}{2} > \frac{- 6}{2}

Simplify

\frac{2 u}{2} > \frac{- 6}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 6}{2} : - 3

\frac{- 6}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{2}

Cancel

\frac{6}{2} : 3

\frac{6}{2}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 3}{2}

Cancel the common factor:

2

= 3

= - 3

u > - 3

u > - 3

u > - 3

Find the signs of

2 u - \frac{3}{2}

2 u - \frac{3}{2} = 0 : u = \frac{3}{2}

2 u - \frac{3}{2} = 0

Move

\frac{3}{2}

to the right side

2 u - \frac{3}{2} = 0

Add

\frac{3}{2}

to both sides

2 u - \frac{3}{2} + \frac{3}{2} = 0 + \frac{3}{2}

Simplify

2 u = \frac{3}{2}

2 u = \frac{3}{2}

Divide both sides by

2

2 u = \frac{3}{2}

Divide both sides by

2

\frac{2 u}{2} = \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} = \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{\frac{3}{2}}{2} : \frac{3}{2}

\frac{\frac{3}{2}}{2}

\frac{3}{2} = \frac{3}{2}

\frac{3}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{2}

= \frac{\frac{3}{2}}{2}

Apply the fraction rule:

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

= \frac{3}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{3}{2}

u = \frac{3}{2}

u = \frac{3}{2}

u = \frac{3}{2}

2 u - \frac{3}{2} < 0 : u < \frac{3}{2}

2 u - \frac{3}{2} < 0

Move

\frac{3}{2}

to the right side

2 u - \frac{3}{2} < 0

Add

\frac{3}{2}

to both sides

2 u - \frac{3}{2} + \frac{3}{2} < 0 + \frac{3}{2}

Simplify

2 u < \frac{3}{2}

2 u < \frac{3}{2}

Divide both sides by

2

2 u < \frac{3}{2}

Divide both sides by

2

\frac{2 u}{2} < \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} < \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{\frac{3}{2}}{2} : \frac{3}{2}

\frac{\frac{3}{2}}{2}

\frac{3}{2} = \frac{3}{2}

\frac{3}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{2}

= \frac{\frac{3}{2}}{2}

Apply the fraction rule:

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

= \frac{3}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{3}{2}

u < \frac{3}{2}

u < \frac{3}{2}

u < \frac{3}{2}

2 u - \frac{3}{2} > 0 : u > \frac{3}{2}

2 u - \frac{3}{2} > 0

Move

\frac{3}{2}

to the right side

2 u - \frac{3}{2} > 0

Add

\frac{3}{2}

to both sides

2 u - \frac{3}{2} + \frac{3}{2} > 0 + \frac{3}{2}

Simplify

2 u > \frac{3}{2}

2 u > \frac{3}{2}

Divide both sides by

2

2 u > \frac{3}{2}

Divide both sides by

2

\frac{2 u}{2} > \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} > \frac{\frac{3}{2}}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{\frac{3}{2}}{2} : \frac{3}{2}

\frac{\frac{3}{2}}{2}

\frac{3}{2} = \frac{3}{2}

\frac{3}{2}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{2}

= \frac{\frac{3}{2}}{2}

Apply the fraction rule:

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

= \frac{3}{2 2}

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{3}{2}

u > \frac{3}{2}

u > \frac{3}{2}

u > \frac{3}{2}

Summarize in a table:

2 u + 6 2 u - \frac{3}{2} (2 u + 6) (2 u - \frac{3}{2}) u < - 3 - - + u = - 3 0 - 0 - 3 < u < \frac{3}{2} + - - u = \frac{3}{2} + 00 u > \frac{3}{2} + + +

Identify the intervals that satisfy the required condition:

> 0

u < - 3 or u > \frac{3}{2}

u < - 3 or u > \frac{3}{2}

Summarize in a table:

2 (u + \frac{3}{4})^{2} - \frac{27}{8} 2 (u + \frac{3}{4})^{2} - \frac{27}{8} u < - 3 + + u = - 3 00 - 3 < u < \frac{3}{2} - - u = \frac{3}{2} 00 u > \frac{3}{2} + +

Identify the intervals that satisfy the required condition:

\leq 0

u = - 3 or - 3 < u < \frac{3}{2} or u = \frac{3}{2}

Merge Overlapping Intervals

- 3 \leq u < \frac{3}{2} or u = \frac{3}{2}

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

u = - 3

- 3 < u < \frac{3}{2}

- 3 \leq u < \frac{3}{2}

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

- 3 \leq u < \frac{3}{2}

u = \frac{3}{2}

- 3 \leq u \leq \frac{3}{2}

- 3 \leq u \leq \frac{3}{2}

- 3 \leq u \leq \frac{3}{2}

- 3 \leq u \leq \frac{3}{2}

Substitute back

u = cos (x)

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

a \leq u \leq b

then

a \leq u and u \leq b

- 3 \leq cos (x) and cos (x) \leq \frac{3}{2}

- 3 \leq cos (x) :

True for all

x \in R

- 3 \leq cos (x)

Switch sides

cos (x) \geq - 3

Range of

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

Function range definition

The range of the basic

cos

function is

- 1 \leq cos (x) \leq 1

- 1 \leq cos (x) \leq 1

cos (x) \geq - 3 and - 1 \leq cos (x) \leq 1 : - 1 \leq cos (x) \leq 1

Let

y = cos (x)

Combine the intervals

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

Merge Overlapping Intervals

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

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

y \geq - 3

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

cos (x) \leq \frac{3}{2} : \frac{π}{6} + 2 πn \leq x \leq \frac{11 π}{6} + 2 πn

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

For

cos (x) \leq a

, if

- 1 < a < 1

then

arccos (a) + 2 πn \leq x \leq 2 π - arccos (a) + 2 πn

arccos (\frac{3}{2}) + 2 πn \leq x \leq 2 π - arccos (\frac{3}{2}) + 2 πn

Simplify

arccos (\frac{3}{2}) : \frac{π}{6}

arccos (\frac{3}{2})

Use the following trivial identity:

arccos (\frac{3}{2}) = \frac{π}{6}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{π}{6}

Simplify

2 π - arccos (\frac{3}{2}) : \frac{11 π}{6}

2 π - arccos (\frac{3}{2})

Use the following trivial identity:

arccos (\frac{3}{2}) = \frac{π}{6}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= 2 π - \frac{π}{6}

Simplify

2 π - \frac{π}{6}

Convert element to fraction:

2 π = \frac{2 π 6}{6}

= \frac{2 π 6}{6} - \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{2 π 6 - π}{6}

2 π 6 - π = 11 π

2 π 6 - π

Multiply the numbers:

2 \cdot 6 = 12

= 12 π - π

Add similar elements:

12 π - π = 11 π

= 11 π

= \frac{11 π}{6}

= \frac{11 π}{6}

\frac{π}{6} + 2 πn \leq x \leq \frac{11 π}{6} + 2 πn

Combine the intervals

True for all x \in R and \frac{π}{6} + 2 πn \leq x \leq \frac{11 π}{6} + 2 πn

Merge Overlapping Intervals

\frac{π}{6} + 2 πn \leq x \leq \frac{11 π}{6} + 2 πn

Popular Examples

sin^2(x)< 3/4

sin^{2} (x) < \frac{3}{4}

sin(x)>= (sqrt(2))/2

sin (x) \geq \frac{2}{2}

tan(x)>(sqrt(3))/3

tan (x) > \frac{3}{3}

4cos^2(3x-1)>0

4 cos^{2} (3 x - 1) > 0

sin^2(x)<0

sin^{2} (x) < 0