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2cos^2(x)<= sqrt(3)sin(x)

Solution

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

Solution

arcsin (\frac{19 - 3}{4}) + 2 πn \leq x \leq π - arcsin (\frac{19 - 3}{4}) + 2 πn

Interval Notation

[arcsin (\frac{19 - 3}{4}) + 2 πn, π - arcsin (\frac{19 - 3}{4}) + 2 πn]

Decimal Notation

0.71645 \dots + 2 πn \leq x \leq 2.42514 \dots + 2 πn

Solution steps

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

Move

3 sin (x)

to the left side

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

Subtract

3 sin (x)

from both sides

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

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

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

Use the following identity:

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

Therefore

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

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

Let:

u = sin (x)

2 (1 - u^{2}) - 3 u \leq 0

2 (1 - u^{2}) - 3 u \leq 0 : u \leq \frac{- 19 - 3}{4} or u \geq \frac{19 - 3}{4}

2 (1 - u^{2}) - 3 u \leq 0

Factor

2 (1 - u^{2}) - 3 u : - 2 (u + \frac{3}{4})^{2} + \frac{19}{8}

2 (1 - u^{2}) - 3 u

2 (1 - u^{2}) = - 2 (u + 1) (u - 1)

2 (1 - u^{2})

Factor

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

- u^{2} + 1

Factor out common term

- 1

= - (u^{2} - 1)

Factor

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

u^{2} - 1

Rewrite

1

1^{2}

= u^{2} - 1^{2}

Apply Difference of Two Squares Formula:

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

u^{2} - 1^{2} = (u + 1) (u - 1)

= (u + 1) (u - 1)

= - (u + 1) (u - 1)

= - 2 (u + 1) (u - 1)

= - 2 (u + 1) (u - 1) - 3 u

Expand

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

- 2 (u + 1) (u - 1) - 3 u

Expand

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

Expand

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

(u + 1) (u - 1)

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = u, b = 1

= u^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= u^{2} - 1

= - 2 (u^{2} - 1)

Expand

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

- 2 (u^{2} - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = u^{2}, c = 1

= - 2 u^{2} - (- 2) \cdot 1

Apply minus-plus rules

- (- a) = a

= - 2 u^{2} + 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= - 2 u^{2} + 2

= - 2 u^{2} + 2

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

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

Factor

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

Factor out common term

- 1

Multiply the numbers:

1 \cdot 1 \cdot 2 = 2

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

Complete the square

- (2 u^{2} + 3 u - 2) : - 2 (u + \frac{3}{4})^{2} + \frac{19}{8}

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

Write in the standard form

a x^{2} + b x + c

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

Write

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

in the form:

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

Factor out

- 2

- 2 (u^{2} + \frac{3 u}{2} - 1)

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} - 1 + (\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} - 1 - (\frac{3}{4})^{2}

Simplify

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8}

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} \leq 0

Factor

2 u^{2} + 3 u - 2 : (u - \frac{- 3 + 19}{4}) (u - \frac{- 3 - 19}{4})

2 u^{2} + 3 u - 2

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 - 2 = 0 : u = \frac{- 3 + 19}{4}, u = \frac{- 3 - 19}{4}

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

(3)^{2} - 4 \cdot 2 (- 2) = 19

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

Apply rule

- (- a) = a

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

(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 2 = 16

4 \cdot 2 \cdot 2

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 16

= 3 + 16

Add the numbers:

3 + 16 = 19

= 19

u_{1, 2} = \frac{- 3 \pm 19}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 3 + 19}{2 \cdot 2}, u_{2} = \frac{- 3 - 19}{2 \cdot 2}

u = \frac{- 3 + 19}{2 \cdot 2} : \frac{- 3 + 19}{4}

\frac{- 3 + 19}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 3 + 19}{4}

u = \frac{- 3 - 19}{2 \cdot 2} : \frac{- 3 - 19}{4}

\frac{- 3 - 19}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 3 - 19}{4}

The solutions to the quadratic equation are:

u = \frac{- 3 + 19}{4}, u = \frac{- 3 - 19}{4}

2 u^{2} + 3 u - 2 = (u - \frac{- 3 + 19}{4}) (u - \frac{- 3 - 19}{4})

= - 2 (u + \frac{3}{4})^{2} + \frac{19}{8}

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} \leq 0

Identify the intervals

Find the signs of the factors of

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8}

Find the signs of

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8}

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} = 0 : u = \frac{- 19 - 3}{4} or u = \frac{19 - 3}{4}

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} = 0

Move

\frac{19}{8}

to the right side

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} = 0

Subtract

\frac{19}{8}

from both sides

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} - \frac{19}{8} = 0 - \frac{19}{8}

Simplify

- 2 (u + \frac{3}{4})^{2} = - \frac{19}{8}

- 2 (u + \frac{3}{4})^{2} = - \frac{19}{8}

Divide both sides by

- 2

- 2 (u + \frac{3}{4})^{2} = - \frac{19}{8}

Divide both sides by

- 2

\frac{- 2 ( u + \frac{3}{4} ) ^{2}}{- 2} = \frac{- \frac{19}{8}}{- 2}

Simplify

\frac{- 2 ( u + \frac{3}{4} ) ^{2}}{- 2} = \frac{- \frac{19}{8}}{- 2}

Simplify

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

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

Apply the fraction rule:

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

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

Divide the numbers:

\frac{2}{2} = 1

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

Simplify

\frac{- \frac{19}{8}}{- 2} : \frac{19}{16}

\frac{- \frac{19}{8}}{- 2}

Apply the fraction rule:

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

= \frac{\frac{19}{8}}{2}

Apply the fraction rule:

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

= \frac{19}{8 \cdot 2}

Multiply the numbers:

8 \cdot 2 = 16

= \frac{19}{16}

(u + \frac{3}{4})^{2} = \frac{19}{16}

(u + \frac{3}{4})^{2} = \frac{19}{16}

(u + \frac{3}{4})^{2} = \frac{19}{16}

For

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

the solutions are

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

Solve

u + \frac{3}{4} = \frac{19}{16} : u = \frac{19 - 3}{4}

u + \frac{3}{4} = \frac{19}{16}

Apply radical rule:

assuming

a \geq 0, b \geq 0

u + \frac{3}{4} = \frac{19}{16}

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

u + \frac{3}{4} = \frac{19}{4}

Move

\frac{3}{4}

to the right side

u + \frac{3}{4} = \frac{19}{4}

Subtract

\frac{3}{4}

from both sides

u + \frac{3}{4} - \frac{3}{4} = \frac{19}{4} - \frac{3}{4}

Simplify

u + \frac{3}{4} - \frac{3}{4} = \frac{19}{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{19}{4} - \frac{3}{4} : \frac{19 - 3}{4}

\frac{19}{4} - \frac{3}{4}

Apply rule

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

= \frac{19 - 3}{4}

u = \frac{19 - 3}{4}

u = \frac{19 - 3}{4}

u = \frac{19 - 3}{4}

Solve

u + \frac{3}{4} = - \frac{19}{16} : u = \frac{- 19 - 3}{4}

u + \frac{3}{4} = - \frac{19}{16}

Simplify

\frac{19}{16} : \frac{19}{4}

\frac{19}{16}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{19}{16}

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

= \frac{19}{4}

u + \frac{3}{4} = - \frac{19}{4}

Move

\frac{3}{4}

to the right side

u + \frac{3}{4} = - \frac{19}{4}

Subtract

\frac{3}{4}

from both sides

u + \frac{3}{4} - \frac{3}{4} = - \frac{19}{4} - \frac{3}{4}

Simplify

u + \frac{3}{4} - \frac{3}{4} = - \frac{19}{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{19}{4} - \frac{3}{4} : \frac{- 19 - 3}{4}

- \frac{19}{4} - \frac{3}{4}

Apply rule

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

= \frac{- 19 - 3}{4}

u = \frac{- 19 - 3}{4}

u = \frac{- 19 - 3}{4}

u = \frac{- 19 - 3}{4}

The solutions to the quadratic equation are:

u = \frac{19 - 3}{4}, u = \frac{- 19 - 3}{4}

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} < 0 : u < - \frac{3 + 19}{4} or u > - \frac{3 - 19}{4}

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} < 0

Factor

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} : - 2 (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4})

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8}

Factor out common term

- 1

= - 2 (u + \frac{3}{4})^{2} - \frac{19}{8}

Factor

2 (u + \frac{3}{4})^{2} - \frac{19}{8} : (2 (u + \frac{3}{4}) + \frac{19}{8}) (2 (u + \frac{3}{4}) - \frac{19}{8})

2 (u + \frac{3}{4})^{2} - \frac{19}{8}

Rewrite

2 (u + \frac{3}{4})^{2} - \frac{19}{8}

(2 (u + \frac{3}{4}))^{2} - (\frac{19}{8})^{2}

2 (u + \frac{3}{4})^{2} - \frac{19}{8}

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} (u + \frac{3}{4})^{2} - \frac{19}{8}

Apply radical rule:

a = (a)^{2}

\frac{19}{8} = (\frac{19}{8})^{2}

= (2)^{2} (u + \frac{3}{4})^{2} - (\frac{19}{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{19}{8})^{2}

= (2 (u + \frac{3}{4}))^{2} - (\frac{19}{8})^{2}

Apply Difference of Two Squares Formula:

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

(2 (u + \frac{3}{4}))^{2} - (\frac{19}{8})^{2} = (2 (u + \frac{3}{4}) + \frac{19}{8}) (2 (u + \frac{3}{4}) - \frac{19}{8})

= (2 (u + \frac{3}{4}) + \frac{19}{8}) (2 (u + \frac{3}{4}) - \frac{19}{8})

= - (2 (u + \frac{3}{4}) + \frac{19}{8}) (2 (u + \frac{3}{4}) - \frac{19}{8})

Refine

= - (2 (u + \frac{3}{4}) + \frac{19}{2 2}) (2 (u + \frac{3}{4}) - \frac{19}{2 2})

Expand

2 (u + \frac{3}{4}) + \frac{19}{2 2} : 2 u + \frac{3 + 19}{2 2}

2 (u + \frac{3}{4}) + \frac{19}{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{19}{2 2}

Combine the fractions

\frac{3}{2 2} + \frac{19}{2 2} : \frac{3 + 19}{2 2}

Apply rule

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

= \frac{3 + 19}{2 2}

= 2 u + \frac{3 + 19}{2 2}

= - (2 u + \frac{3 + 19}{2 2}) (2 (u + \frac{3}{4}) - \frac{19}{2 2})

Expand

2 (u + \frac{3}{4}) - \frac{19}{2 2} : 2 u + \frac{3 - 19}{2 2}

2 (u + \frac{3}{4}) - \frac{19}{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{19}{2 2}

Combine the fractions

\frac{3}{2 2} - \frac{19}{2 2} : \frac{3 - 19}{2 2}

Apply rule

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

= \frac{3 - 19}{2 2}

= 2 u + \frac{3 - 19}{2 2}

= - (2 u + \frac{3 + 19}{2 2}) (2 u + \frac{3 - 19}{2 2})

\frac{3 + 19}{2 2} = \frac{2 ( 3 + 19 )}{4}

\frac{3 + 19}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{( 3 + 19 ) 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Refine

= 4

= \frac{2 ( 3 + 19 )}{4}

= - (2 u + \frac{2 ( 3 + 19 )}{4}) (2 u + \frac{3 - 19}{2 2})

\frac{3 - 19}{2 2} = \frac{2 ( 3 - 19 )}{4}

\frac{3 - 19}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{( 3 - 19 ) 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Refine

= 4

= \frac{2 ( 3 - 19 )}{4}

= - (2 u + \frac{2 ( 3 + 19 )}{4}) (2 u + \frac{2 ( 3 - 19 )}{4})

Factor

2 u + \frac{2 ( 3 + 19 )}{4} : 2 (u + \frac{3 + 19}{4})

2 u + \frac{2 ( 3 + 19 )}{4}

Factor out common term

2

= 2 (u + \frac{3 + 19}{4})

= - 2 (u + \frac{3 + 19}{4}) (2 u + \frac{2 ( 3 - 19 )}{4})

Factor

2 u + \frac{2 ( 3 - 19 )}{4} : 2 (u + \frac{3 - 19}{4})

2 u + \frac{2 ( 3 - 19 )}{4}

Factor out common term

2

= 2 (u + \frac{3 - 19}{4})

= - 2 (u + \frac{3 + 19}{4}) 2 (u + \frac{3 - 19}{4})

Refine

= - 2 (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4})

- 2 (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4}) < 0

Multiply both sides by

- 1

(reverse the inequality)

(- 2 (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4})) (- 1) > 0 \cdot (- 1)

Simplify

2 (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4}) > 0

Divide both sides by

2

\frac{2 ( u + \frac{3 + 19}{4} ) ( u + \frac{3 - 19}{4} )}{2} > \frac{0}{2}

Simplify

(u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4}) > 0

Identify the intervals

Find the signs of the factors of

(u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4})

Find the signs of

u + \frac{3 + 19}{4}

u + \frac{3 + 19}{4} = 0 : u = - \frac{3 + 19}{4}

u + \frac{3 + 19}{4} = 0

Move

\frac{3 + 19}{4}

to the right side

u + \frac{3 + 19}{4} = 0

Subtract

\frac{3 + 19}{4}

from both sides

u + \frac{3 + 19}{4} - \frac{3 + 19}{4} = 0 - \frac{3 + 19}{4}

Simplify

u = - \frac{3 + 19}{4}

u = - \frac{3 + 19}{4}

u + \frac{3 + 19}{4} < 0 : u < - \frac{3 + 19}{4}

u + \frac{3 + 19}{4} < 0

Move

\frac{3 + 19}{4}

to the right side

u + \frac{3 + 19}{4} < 0

Subtract

\frac{3 + 19}{4}

from both sides

u + \frac{3 + 19}{4} - \frac{3 + 19}{4} < 0 - \frac{3 + 19}{4}

Simplify

u < - \frac{3 + 19}{4}

u < - \frac{3 + 19}{4}

u + \frac{3 + 19}{4} > 0 : u > - \frac{3 + 19}{4}

u + \frac{3 + 19}{4} > 0

Move

\frac{3 + 19}{4}

to the right side

u + \frac{3 + 19}{4} > 0

Subtract

\frac{3 + 19}{4}

from both sides

u + \frac{3 + 19}{4} - \frac{3 + 19}{4} > 0 - \frac{3 + 19}{4}

Simplify

u > - \frac{3 + 19}{4}

u > - \frac{3 + 19}{4}

Find the signs of

u + \frac{3 - 19}{4}

u + \frac{3 - 19}{4} = 0 : u = - \frac{3 - 19}{4}

u + \frac{3 - 19}{4} = 0

Move

\frac{3 - 19}{4}

to the right side

u + \frac{3 - 19}{4} = 0

Subtract

\frac{3 - 19}{4}

from both sides

u + \frac{3 - 19}{4} - \frac{3 - 19}{4} = 0 - \frac{3 - 19}{4}

Simplify

u = - \frac{3 - 19}{4}

u = - \frac{3 - 19}{4}

u + \frac{3 - 19}{4} < 0 : u < - \frac{3 - 19}{4}

u + \frac{3 - 19}{4} < 0

Move

\frac{3 - 19}{4}

to the right side

u + \frac{3 - 19}{4} < 0

Subtract

\frac{3 - 19}{4}

from both sides

u + \frac{3 - 19}{4} - \frac{3 - 19}{4} < 0 - \frac{3 - 19}{4}

Simplify

u < - \frac{3 - 19}{4}

u < - \frac{3 - 19}{4}

u + \frac{3 - 19}{4} > 0 : u > - \frac{3 - 19}{4}

u + \frac{3 - 19}{4} > 0

Move

\frac{3 - 19}{4}

to the right side

u + \frac{3 - 19}{4} > 0

Subtract

\frac{3 - 19}{4}

from both sides

u + \frac{3 - 19}{4} - \frac{3 - 19}{4} > 0 - \frac{3 - 19}{4}

Simplify

u > - \frac{3 - 19}{4}

u > - \frac{3 - 19}{4}

Summarize in a table:

u + \frac{3 + 19}{4} u + \frac{3 - 19}{4} (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4}) u < - \frac{3 + 19}{4} - - + u = - \frac{3 + 19}{4} 0 - 0 - \frac{3 + 19}{4} < u < - \frac{3 - 19}{4} + - - u = - \frac{3 - 19}{4} + 00 u > - \frac{3 - 19}{4} + + +

Identify the intervals that satisfy the required condition:

> 0

u < - \frac{3 + 19}{4} or u > - \frac{3 - 19}{4}

u < - \frac{3 + 19}{4} or u > - \frac{3 - 19}{4}

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} > 0 : - \frac{3 + 19}{4} < u < - \frac{3 - 19}{4}

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} > 0

Factor

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} : - 2 (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4})

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8}

Factor out common term

- 1

= - 2 (u + \frac{3}{4})^{2} - \frac{19}{8}

Factor

2 (u + \frac{3}{4})^{2} - \frac{19}{8} : (2 (u + \frac{3}{4}) + \frac{19}{8}) (2 (u + \frac{3}{4}) - \frac{19}{8})

2 (u + \frac{3}{4})^{2} - \frac{19}{8}

Rewrite

2 (u + \frac{3}{4})^{2} - \frac{19}{8}

(2 (u + \frac{3}{4}))^{2} - (\frac{19}{8})^{2}

2 (u + \frac{3}{4})^{2} - \frac{19}{8}

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

= (2)^{2} (u + \frac{3}{4})^{2} - \frac{19}{8}

Apply radical rule:

a = (a)^{2}

\frac{19}{8} = (\frac{19}{8})^{2}

= (2)^{2} (u + \frac{3}{4})^{2} - (\frac{19}{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{19}{8})^{2}

= (2 (u + \frac{3}{4}))^{2} - (\frac{19}{8})^{2}

Apply Difference of Two Squares Formula:

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

(2 (u + \frac{3}{4}))^{2} - (\frac{19}{8})^{2} = (2 (u + \frac{3}{4}) + \frac{19}{8}) (2 (u + \frac{3}{4}) - \frac{19}{8})

= (2 (u + \frac{3}{4}) + \frac{19}{8}) (2 (u + \frac{3}{4}) - \frac{19}{8})

= - (2 (u + \frac{3}{4}) + \frac{19}{8}) (2 (u + \frac{3}{4}) - \frac{19}{8})

Refine

= - (2 (u + \frac{3}{4}) + \frac{19}{2 2}) (2 (u + \frac{3}{4}) - \frac{19}{2 2})

Expand

2 (u + \frac{3}{4}) + \frac{19}{2 2} : 2 u + \frac{3 + 19}{2 2}

2 (u + \frac{3}{4}) + \frac{19}{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{19}{2 2}

Combine the fractions

\frac{3}{2 2} + \frac{19}{2 2} : \frac{3 + 19}{2 2}

Apply rule

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

= \frac{3 + 19}{2 2}

= 2 u + \frac{3 + 19}{2 2}

= - (2 u + \frac{3 + 19}{2 2}) (2 (u + \frac{3}{4}) - \frac{19}{2 2})

Expand

2 (u + \frac{3}{4}) - \frac{19}{2 2} : 2 u + \frac{3 - 19}{2 2}

2 (u + \frac{3}{4}) - \frac{19}{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{19}{2 2}

Combine the fractions

\frac{3}{2 2} - \frac{19}{2 2} : \frac{3 - 19}{2 2}

Apply rule

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

= \frac{3 - 19}{2 2}

= 2 u + \frac{3 - 19}{2 2}

= - (2 u + \frac{3 + 19}{2 2}) (2 u + \frac{3 - 19}{2 2})

\frac{3 + 19}{2 2} = \frac{2 ( 3 + 19 )}{4}

\frac{3 + 19}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{( 3 + 19 ) 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Refine

= 4

= \frac{2 ( 3 + 19 )}{4}

= - (2 u + \frac{2 ( 3 + 19 )}{4}) (2 u + \frac{3 - 19}{2 2})

\frac{3 - 19}{2 2} = \frac{2 ( 3 - 19 )}{4}

\frac{3 - 19}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{( 3 - 19 ) 2}{2 2 2}

222 = 4

222

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

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

Add similar elements:

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Refine

= 4

= \frac{2 ( 3 - 19 )}{4}

= - (2 u + \frac{2 ( 3 + 19 )}{4}) (2 u + \frac{2 ( 3 - 19 )}{4})

Factor

2 u + \frac{2 ( 3 + 19 )}{4} : 2 (u + \frac{3 + 19}{4})

2 u + \frac{2 ( 3 + 19 )}{4}

Factor out common term

2

= 2 (u + \frac{3 + 19}{4})

= - 2 (u + \frac{3 + 19}{4}) (2 u + \frac{2 ( 3 - 19 )}{4})

Factor

2 u + \frac{2 ( 3 - 19 )}{4} : 2 (u + \frac{3 - 19}{4})

2 u + \frac{2 ( 3 - 19 )}{4}

Factor out common term

2

= 2 (u + \frac{3 - 19}{4})

= - 2 (u + \frac{3 + 19}{4}) 2 (u + \frac{3 - 19}{4})

Refine

= - 2 (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4})

- 2 (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4}) > 0

Multiply both sides by

- 1

(reverse the inequality)

(- 2 (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4})) (- 1) < 0 \cdot (- 1)

Simplify

2 (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4}) < 0

Divide both sides by

2

\frac{2 ( u + \frac{3 + 19}{4} ) ( u + \frac{3 - 19}{4} )}{2} < \frac{0}{2}

Simplify

(u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4}) < 0

Identify the intervals

Find the signs of the factors of

(u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4})

Find the signs of

u + \frac{3 + 19}{4}

u + \frac{3 + 19}{4} = 0 : u = - \frac{3 + 19}{4}

u + \frac{3 + 19}{4} = 0

Move

\frac{3 + 19}{4}

to the right side

u + \frac{3 + 19}{4} = 0

Subtract

\frac{3 + 19}{4}

from both sides

u + \frac{3 + 19}{4} - \frac{3 + 19}{4} = 0 - \frac{3 + 19}{4}

Simplify

u = - \frac{3 + 19}{4}

u = - \frac{3 + 19}{4}

u + \frac{3 + 19}{4} < 0 : u < - \frac{3 + 19}{4}

u + \frac{3 + 19}{4} < 0

Move

\frac{3 + 19}{4}

to the right side

u + \frac{3 + 19}{4} < 0

Subtract

\frac{3 + 19}{4}

from both sides

u + \frac{3 + 19}{4} - \frac{3 + 19}{4} < 0 - \frac{3 + 19}{4}

Simplify

u < - \frac{3 + 19}{4}

u < - \frac{3 + 19}{4}

u + \frac{3 + 19}{4} > 0 : u > - \frac{3 + 19}{4}

u + \frac{3 + 19}{4} > 0

Move

\frac{3 + 19}{4}

to the right side

u + \frac{3 + 19}{4} > 0

Subtract

\frac{3 + 19}{4}

from both sides

u + \frac{3 + 19}{4} - \frac{3 + 19}{4} > 0 - \frac{3 + 19}{4}

Simplify

u > - \frac{3 + 19}{4}

u > - \frac{3 + 19}{4}

Find the signs of

u + \frac{3 - 19}{4}

u + \frac{3 - 19}{4} = 0 : u = - \frac{3 - 19}{4}

u + \frac{3 - 19}{4} = 0

Move

\frac{3 - 19}{4}

to the right side

u + \frac{3 - 19}{4} = 0

Subtract

\frac{3 - 19}{4}

from both sides

u + \frac{3 - 19}{4} - \frac{3 - 19}{4} = 0 - \frac{3 - 19}{4}

Simplify

u = - \frac{3 - 19}{4}

u = - \frac{3 - 19}{4}

u + \frac{3 - 19}{4} < 0 : u < - \frac{3 - 19}{4}

u + \frac{3 - 19}{4} < 0

Move

\frac{3 - 19}{4}

to the right side

u + \frac{3 - 19}{4} < 0

Subtract

\frac{3 - 19}{4}

from both sides

u + \frac{3 - 19}{4} - \frac{3 - 19}{4} < 0 - \frac{3 - 19}{4}

Simplify

u < - \frac{3 - 19}{4}

u < - \frac{3 - 19}{4}

u + \frac{3 - 19}{4} > 0 : u > - \frac{3 - 19}{4}

u + \frac{3 - 19}{4} > 0

Move

\frac{3 - 19}{4}

to the right side

u + \frac{3 - 19}{4} > 0

Subtract

\frac{3 - 19}{4}

from both sides

u + \frac{3 - 19}{4} - \frac{3 - 19}{4} > 0 - \frac{3 - 19}{4}

Simplify

u > - \frac{3 - 19}{4}

u > - \frac{3 - 19}{4}

Summarize in a table:

u + \frac{3 + 19}{4} u + \frac{3 - 19}{4} (u + \frac{3 + 19}{4}) (u + \frac{3 - 19}{4}) u < - \frac{3 + 19}{4} - - + u = - \frac{3 + 19}{4} 0 - 0 - \frac{3 + 19}{4} < u < - \frac{3 - 19}{4} + - - u = - \frac{3 - 19}{4} + 00 u > - \frac{3 - 19}{4} + + +

Identify the intervals that satisfy the required condition:

< 0

- \frac{3 + 19}{4} < u < - \frac{3 - 19}{4}

- \frac{3 + 19}{4} < u < - \frac{3 - 19}{4}

Summarize in a table:

- 2 (u + \frac{3}{4})^{2} + \frac{19}{8} - 2 (u + \frac{3}{4})^{2} + \frac{19}{8} u < \frac{- 19 - 3}{4} - - u = \frac{- 19 - 3}{4} 00 \frac{- 19 - 3}{4} < u < \frac{19 - 3}{4} + + u = \frac{19 - 3}{4} 00 u > \frac{19 - 3}{4} - -

Identify the intervals that satisfy the required condition:

\leq 0

u < \frac{- 19 - 3}{4} or u = \frac{- 19 - 3}{4} or u = \frac{19 - 3}{4} or u > \frac{19 - 3}{4}

Merge Overlapping Intervals

u \leq \frac{- 19 - 3}{4} or u = \frac{19 - 3}{4} or u > \frac{19 - 3}{4}

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

u < \frac{- 19 - 3}{4}

u = \frac{- 19 - 3}{4}

u \leq \frac{- 19 - 3}{4}

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

u \leq \frac{- 19 - 3}{4}

u = \frac{19 - 3}{4}

u \leq \frac{- 19 - 3}{4} or u = \frac{19 - 3}{4}

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

u \leq \frac{- 19 - 3}{4} or u = \frac{19 - 3}{4}

u > \frac{19 - 3}{4}

u \leq \frac{- 19 - 3}{4} or u \geq \frac{19 - 3}{4}

u \leq \frac{- 19 - 3}{4} or u \geq \frac{19 - 3}{4}

u \leq \frac{- 19 - 3}{4} or u \geq \frac{19 - 3}{4}

u \leq \frac{- 19 - 3}{4} or u \geq \frac{19 - 3}{4}

Substitute back

u = sin (x)

sin (x) \leq \frac{- 19 - 3}{4} or sin (x) \geq \frac{19 - 3}{4}

sin (x) \leq \frac{- 19 - 3}{4} :

False for all

x \in R

sin (x) \leq \frac{- 19 - 3}{4}

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 \frac{- 19 - 3}{4} and - 1 \leq sin (x) \leq 1 :

False

Let

y = sin (x)

Combine the intervals

y \leq \frac{- 19 - 3}{4} and - 1 \leq y \leq 1

Merge Overlapping Intervals

y \leq \frac{- 19 - 3}{4} and - 1 \leq y \leq 1

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

y \leq \frac{- 19 - 3}{4}

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 \frac{19 - 3}{4} : arcsin (\frac{19 - 3}{4}) + 2 πn \leq x \leq π - arcsin (\frac{19 - 3}{4}) + 2 πn

sin (x) \geq \frac{19 - 3}{4}

For

sin (x) \geq a

, if

- 1 < a < 1

then

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

arcsin (\frac{19 - 3}{4}) + 2 πn \leq x \leq π - arcsin (\frac{19 - 3}{4}) + 2 πn

Combine the intervals

False for all x \in R or arcsin (\frac{19 - 3}{4}) + 2 πn \leq x \leq π - arcsin (\frac{19 - 3}{4}) + 2 πn

Merge Overlapping Intervals

arcsin (\frac{19 - 3}{4}) + 2 πn \leq x \leq π - arcsin (\frac{19 - 3}{4}) + 2 πn

Popular Examples

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tan^2(x)>3

tan^{2} (x) > 3

sin(x+pi/3)> 1/2

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sqrt(3)-tan(x)>= 0

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