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sqrt(2)sin^2(x)+(1-sqrt(2))sin(x)-1<= 0

Solution

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

Solution

- \frac{π}{4} + 2 πn \leq x \leq \frac{5 π}{4} + 2 πn

Interval Notation

[- \frac{π}{4} + 2 πn, \frac{5 π}{4} + 2 πn]

Decimal Notation

- 0.78539 \dots + 2 πn \leq x \leq 3.92699 \dots + 2 πn

Solution steps

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

Let:

u = sin (x)

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

2 u^{2} + (1 - 2) u - 1 \leq 0 : - \frac{2}{2} \leq u \leq 1

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

Factor

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

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

Factor

1 - 2 : - (2 - 1)

1 - 2

Factor out common term

- 1

= - (2 - 1)

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

Complete the square

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

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

Write in the standard form

a x^{2} + b x + c

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

Write

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

in the form:

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

Factor out

2

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

Expand

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

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

Apply rule

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

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

Expand

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

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

Expand

u (- 2 + 1) : - 2 u + u

u (- 2 + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = u, b = - 2, c = 1

= u (- 2) + u \cdot 1

Apply minus-plus rules

+ (- a) = - a

= - 2 u + 1 \cdot u

Multiply:

1 \cdot u = u

= - 2 u + u

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

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

Apply the fraction rule:

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

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

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

Cancel

\frac{2 u ^{2}}{2} : u^{2}

\frac{2 u ^{2}}{2}

Cancel the common factor:

2

= u^{2}

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

Cancel

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

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

2 a = - 1 + \frac{1}{2} : a = \frac{- 2 + 1}{2 2}

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 a}{2} : a

\frac{2 a}{2}

Divide the numbers:

\frac{2}{2} = 1

= a

Simplify

- \frac{1}{2} + \frac{\frac{1}{2}}{2} : \frac{- 2 + 1}{2 2}

- \frac{1}{2} + \frac{\frac{1}{2}}{2}

Apply rule

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

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

Join

- 1 + \frac{1}{2} : \frac{- 2 + 1}{2}

- 1 + \frac{1}{2}

Convert element to fraction:

1 = \frac{1 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot 2 = 2

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

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

Apply the fraction rule:

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

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

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

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

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

Add and subtract

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

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

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

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

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

Simplify

2 u^{2} - 2 u + u - 1

2 u^{2} - 2 u + u - 1 \leq 0

2 u^{2} - 2 u + u - 1 \leq 0

Identify the intervals

Find the signs of the factors of

2 u^{2} - 2 u + u - 1

Find the signs of

2 u^{2} - 2 u + u - 1

2 u^{2} - 2 u + u - 1 = 0 : u = - \frac{2}{2} or u = 1

2 u^{2} - 2 u + u - 1 = 0

Write in the standard form

a x^{2} + b x + c = 0

2 u^{2} + (- 2 + 1) u - 1 = 0

Solve with the quadratic formula

2 u^{2} + (- 2 + 1) u - 1 = 0

Quadratic Equation Formula:

For

a = 2, b = - 2 + 1, c = - 1

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

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

(- 2 + 1)^{2} - 42 (- 1) = 2 + 1

(- 2 + 1)^{2} - 42 (- 1)

Apply rule

- (- a) = a

= (- 2 + 1)^{2} + 42 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= (1 - 2)^{2} + 42

Expand

(- 2 + 1)^{2} + 42 : 3 + 22

(- 2 + 1)^{2} + 42

(- 2 + 1)^{2} : 3 - 22

Apply Perfect Square Formula:

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

a = - 2, b = 1

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

Simplify

(- 2)^{2} + 2 (- 2) \cdot 1 + 1^{2} : 3 - 22

(- 2)^{2} + 2 (- 2) \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

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

Remove parentheses:

(- a) = - a

= (- 2)^{2} - 22 \cdot 1 + 1

(- 2)^{2} = 2

(- 2)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= (2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

= 2^{\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

= 2

22 \cdot 1 = 22

22 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 22

= 2 - 22 + 1

Add the numbers:

2 + 1 = 3

= 3 - 22

= 3 - 22

= 3 - 22 + 42

Add similar elements:

- 22 + 42 = 22

= 3 + 22

= 3 + 22

= 2 + 22 + 1

= (2)^{2} + 22 + (1)^{2}

1 = 1

1

Apply rule

1 = 1

= 1

= (2)^{2} + 22 + 1^{2}

22 \cdot 1 = 22

22 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 22

= (2)^{2} + 22 \cdot 1 + 1^{2}

Apply Perfect Square Formula:

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

(2)^{2} + 22 \cdot 1 + 1^{2} = (2 + 1)^{2}

= (2 + 1)^{2}

Apply radical rule:

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

= 2 + 1

u_{1, 2} = \frac{- ( - 2 + 1 ) \pm ( 2 + 1 )}{2 2}

Separate the solutions

u_{1} = \frac{- ( - 2 + 1 ) + 2 + 1}{2 2}, u_{2} = \frac{- ( - 2 + 1 ) - ( 2 + 1 )}{2 2}

u = \frac{- ( - 2 + 1 ) + 2 + 1}{2 2} : 1

\frac{- ( - 2 + 1 ) + 2 + 1}{2 2}

Expand

- (- 2 + 1) + 2 + 1 : 22

- (- 2 + 1) + 2 + 1

- (- 2 + 1) : 2 - 1

- (- 2 + 1)

Distribute parentheses

= - (- 2) - (1)

Apply minus-plus rules

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

= 2 - 1

= 2 - 1 + 2 + 1

Simplify

2 - 1 + 2 + 1 : 22

2 - 1 + 2 + 1

Add similar elements:

2 + 2 = 22

= 22 - 1 + 1

- 1 + 1 = 0

= 22

= 22

= \frac{2 2}{2 2}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 2 + 1 ) - ( 2 + 1 )}{2 2} : - \frac{2}{2}

\frac{- ( - 2 + 1 ) - ( 2 + 1 )}{2 2}

Expand

- (- 2 + 1) - (2 + 1) : - 2

- (- 2 + 1) - (2 + 1)

- (- 2 + 1) : 2 - 1

- (- 2 + 1)

Distribute parentheses

= - (- 2) - (1)

Apply minus-plus rules

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

= 2 - 1

= 2 - 1 - (2 + 1)

- (2 + 1) : - 2 - 1

- (2 + 1)

Distribute parentheses

= - (2) - (1)

Apply minus-plus rules

+ (- a) = - a

= - 2 - 1

= 2 - 1 - 2 - 1

Simplify

2 - 1 - 2 - 1 : - 2

2 - 1 - 2 - 1

Add similar elements:

2 - 2 = 0

= - 1 - 1

Subtract the numbers:

- 1 - 1 = - 2

= - 2

= - 2

= \frac{- 2}{2 2}

Apply the fraction rule:

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

= - \frac{2}{2 2}

Divide the numbers:

\frac{2}{2} = 1

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

The solutions to the quadratic equation are:

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

2 u^{2} - 2 u + u - 1 < 0 : - \frac{2}{2} < u < 1

2 u^{2} - 2 u + u - 1 < 0

Factor

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

2 u^{2} - 2 u + u - 1

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

Factor out

2 u

from

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

2 u^{2} - 2 u

Apply exponent rule:

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

2 u^{2} = 2 uu

= 2 uu - 2 u

Factor out common term

2 u

= 2 u (u - 1)

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

Factor out common term

u - 1

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

(u - 1) (2 u + 1) < 0

Identify the intervals

Find the signs of the factors of

(u - 1) (2 u + 1)

Find the signs of

u - 1

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to 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

Add

1

to 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

Add

1

to both sides

u - 1 + 1 > 0 + 1

Simplify

u > 1

u > 1

Find the signs of

2 u + 1

2 u + 1 = 0 : u = - \frac{2}{2}

2 u + 1 = 0

Move

1

to the right side

2 u + 1 = 0

Subtract

1

from both sides

2 u + 1 - 1 = 0 - 1

Simplify

2 u = - 1

2 u = - 1

Divide both sides by

2

2 u = - 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

2 u + 1 < 0 : u < - \frac{2}{2}

2 u + 1 < 0

Move

1

to the right side

2 u + 1 < 0

Subtract

1

from both sides

2 u + 1 - 1 < 0 - 1

Simplify

2 u < - 1

2 u < - 1

Divide both sides by

2

2 u < - 1

Divide both sides by

2

\frac{2 u}{2} < \frac{- 1}{2}

Simplify

\frac{2 u}{2} < \frac{- 1}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u < - \frac{2}{2}

u < - \frac{2}{2}

u < - \frac{2}{2}

2 u + 1 > 0 : u > - \frac{2}{2}

2 u + 1 > 0

Move

1

to the right side

2 u + 1 > 0

Subtract

1

from both sides

2 u + 1 - 1 > 0 - 1

Simplify

2 u > - 1

2 u > - 1

Divide both sides by

2

2 u > - 1

Divide both sides by

2

\frac{2 u}{2} > \frac{- 1}{2}

Simplify

\frac{2 u}{2} > \frac{- 1}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u > - \frac{2}{2}

u > - \frac{2}{2}

u > - \frac{2}{2}

Summarize in a table:

u - 1 2 u + 1 (u - 1) (2 u + 1) u < - \frac{2}{2} - - + u = - \frac{2}{2} - 00 - \frac{2}{2} < u < 1 - + - u = 1 0 + 0 u > 1 + + +

Identify the intervals that satisfy the required condition:

< 0

- \frac{2}{2} < u < 1

- \frac{2}{2} < u < 1

2 u^{2} - 2 u + u - 1 > 0 : u < - \frac{2}{2} or u > 1

2 u^{2} - 2 u + u - 1 > 0

Factor

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

2 u^{2} - 2 u + u - 1

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

Factor out

2 u

from

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

2 u^{2} - 2 u

Apply exponent rule:

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

2 u^{2} = 2 uu

= 2 uu - 2 u

Factor out common term

2 u

= 2 u (u - 1)

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

Factor out common term

u - 1

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

(u - 1) (2 u + 1) > 0

Identify the intervals

Find the signs of the factors of

(u - 1) (2 u + 1)

Find the signs of

u - 1

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to 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

Add

1

to 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

Add

1

to both sides

u - 1 + 1 > 0 + 1

Simplify

u > 1

u > 1

Find the signs of

2 u + 1

2 u + 1 = 0 : u = - \frac{2}{2}

2 u + 1 = 0

Move

1

to the right side

2 u + 1 = 0

Subtract

1

from both sides

2 u + 1 - 1 = 0 - 1

Simplify

2 u = - 1

2 u = - 1

Divide both sides by

2

2 u = - 1

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

u = - \frac{2}{2}

2 u + 1 < 0 : u < - \frac{2}{2}

2 u + 1 < 0

Move

1

to the right side

2 u + 1 < 0

Subtract

1

from both sides

2 u + 1 - 1 < 0 - 1

Simplify

2 u < - 1

2 u < - 1

Divide both sides by

2

2 u < - 1

Divide both sides by

2

\frac{2 u}{2} < \frac{- 1}{2}

Simplify

\frac{2 u}{2} < \frac{- 1}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u < - \frac{2}{2}

u < - \frac{2}{2}

u < - \frac{2}{2}

2 u + 1 > 0 : u > - \frac{2}{2}

2 u + 1 > 0

Move

1

to the right side

2 u + 1 > 0

Subtract

1

from both sides

2 u + 1 - 1 > 0 - 1

Simplify

2 u > - 1

2 u > - 1

Divide both sides by

2

2 u > - 1

Divide both sides by

2

\frac{2 u}{2} > \frac{- 1}{2}

Simplify

\frac{2 u}{2} > \frac{- 1}{2}

Simplify

\frac{2 u}{2} : u

\frac{2 u}{2}

Cancel the common factor:

2

= u

Simplify

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Apply the fraction rule:

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

= - \frac{1}{2}

Rationalize

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

u > - \frac{2}{2}

u > - \frac{2}{2}

u > - \frac{2}{2}

Summarize in a table:

u - 1 2 u + 1 (u - 1) (2 u + 1) u < - \frac{2}{2} - - + u = - \frac{2}{2} - 00 - \frac{2}{2} < u < 1 - + - u = 1 0 + 0 u > 1 + + +

Identify the intervals that satisfy the required condition:

> 0

u < - \frac{2}{2} or u > 1

u < - \frac{2}{2} or u > 1

Summarize in a table:

2 u^{2} - 2 u + u - 1 2 u^{2} - 2 u + u - 1 u < - \frac{2}{2} + + u = - \frac{2}{2} 00 - \frac{2}{2} < u < 1 - - u = 1 00 u > 1 + +

Identify the intervals that satisfy the required condition:

\leq 0

u = - \frac{2}{2} or - \frac{2}{2} < u < 1 or u = 1

Merge Overlapping Intervals

- \frac{2}{2} \leq u < 1 or u = 1

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

u = - \frac{2}{2}

- \frac{2}{2} < u < 1

- \frac{2}{2} \leq u < 1

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

- \frac{2}{2} \leq u < 1

u = 1

- \frac{2}{2} \leq u \leq 1

- \frac{2}{2} \leq u \leq 1

- \frac{2}{2} \leq u \leq 1

- \frac{2}{2} \leq u \leq 1

Substitute back

u = sin (x)

- \frac{2}{2} \leq sin (x) \leq 1

a \leq u \leq b

then

a \leq u and u \leq b

- \frac{2}{2} \leq sin (x) and sin (x) \leq 1

- \frac{2}{2} \leq sin (x) : - \frac{π}{4} + 2 πn \leq x \leq \frac{5 π}{4} + 2 πn

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

Switch sides

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

For

sin (x) \geq a

, if

- 1 < a < 1

then

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

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

Simplify

arcsin (- \frac{2}{2}) : - \frac{π}{4}

arcsin (- \frac{2}{2})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{2}) = - arcsin (\frac{2}{2})

= - arcsin (\frac{2}{2})

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

arcsin (\frac{2}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{4}

= - \frac{π}{4}

Simplify

π - arcsin (- \frac{2}{2}) : \frac{5 π}{4}

π - arcsin (- \frac{2}{2})

arcsin (- \frac{2}{2}) = - \frac{π}{4}

arcsin (- \frac{2}{2})

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{2}) = - arcsin (\frac{2}{2})

= - arcsin (\frac{2}{2})

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

arcsin (\frac{2}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{4}

= - \frac{π}{4}

= π - (- \frac{π}{4})

Simplify

π - (- \frac{π}{4})

Apply rule

- (- a) = a

= π + \frac{π}{4}

Convert element to fraction:

π = \frac{π 4}{4}

= \frac{π 4}{4} + \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 + π}{4}

Add similar elements:

4 π + π = 5 π

= \frac{5 π}{4}

= \frac{5 π}{4}

- \frac{π}{4} + 2 πn \leq x \leq \frac{5 π}{4} + 2 πn

sin (x) \leq 1 :

True for all

x \in R

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

Let

y = sin (x)

Combine the intervals

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

Merge Overlapping Intervals

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

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

y \leq 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

- \frac{π}{4} + 2 πn \leq x \leq \frac{5 π}{4} + 2 πn and True for all x \in R

Merge Overlapping Intervals

- \frac{π}{4} + 2 πn \leq x \leq \frac{5 π}{4} + 2 πn

Popular Examples

cos(x)< 1/(sqrt(2))

cos (x) < \frac{1}{2}

2cos^2(x)>= 1.5

2 cos^{2} (x) \geq 1.5

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

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

sin^2(x)< 3/4

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

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

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