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2sin^4(2x)>= sin^2(2x)

Solution

2 sin^{4} (2 x) \geq sin^{2} (2 x)

Solution

\frac{π}{8} + πn \leq x \leq \frac{3 π}{8} + πn or x = \frac{π}{2} + πn or - \frac{3 π}{8} + πn \leq x \leq - \frac{π}{8} + πn

Interval Notation

[\frac{π}{8} + πn, \frac{3 π}{8} + πn] \cup x = \frac{π}{2} + πn \cup [- \frac{3 π}{8} + πn, - \frac{π}{8} + πn]

Decimal Notation

0.39269 \dots + πn \leq x \leq 1.17809 \dots + πn or x = 1.57079 \dots + πn or - 1.17809 \dots + πn \leq x \leq - 0.39269 \dots + πn

Solution steps

2 sin^{4} (2 x) \geq sin^{2} (2 x)

Let:

u = sin (2 x)

2 u^{4} \geq u^{2}

2 u^{4} \geq u^{2} : u \leq - \frac{2}{2} or u = 0 or u \geq \frac{2}{2}

2 u^{4} \geq u^{2}

Rewrite in standard form

2 u^{4} \geq u^{2}

Subtract

u^{2}

from both sides

2 u^{4} - u^{2} \geq u^{2} - u^{2}

Simplify

2 u^{4} - u^{2} \geq 0

2 u^{4} - u^{2} \geq 0

Factor

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

2 u^{4} - u^{2}

Factor out common term

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

2 u^{4} - u^{2}

Apply exponent rule:

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

u^{4} = u^{2} u^{2}

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

Factor out common term

u^{2}

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

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

Factor

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

2 u^{2} - 1

Rewrite

2 u^{2} - 1

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

2 u^{2} - 1

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Rewrite

1

1^{2}

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

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

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

Identify the intervals

Find the signs of the factors of

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

Find the signs of

u^{2}

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

u^{2} > 0 : u < 0 or u > 0

u^{2} > 0

For

u^{n} > 0

, if

n

is even then

u < 0

u > 0

u < 0 or u > 0

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}

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

Add

1

to 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}

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}

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

Add

1

to 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}

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}

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

Add

1

to 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}

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}

u > \frac{2}{2}

u > \frac{2}{2}

u > \frac{2}{2}

Summarize in a table:

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

Identify the intervals that satisfy the required condition:

\geq 0

u < - \frac{2}{2} or u = - \frac{2}{2} or u = 0 or u = \frac{2}{2} or u > \frac{2}{2}

Merge Overlapping Intervals

u \leq - \frac{2}{2} or u = 0 or u = \frac{2}{2} or u > \frac{2}{2}

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

u < - \frac{2}{2}

u = - \frac{2}{2}

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

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

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

u = 0

u \leq - \frac{2}{2} or u = 0

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

u \leq - \frac{2}{2} or u = 0

u = \frac{2}{2}

u \leq - \frac{2}{2} or u = 0 or u = \frac{2}{2}

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

u \leq - \frac{2}{2} or u = 0 or u = \frac{2}{2}

u > \frac{2}{2}

u \leq - \frac{2}{2} or u = 0 or u \geq \frac{2}{2}

u \leq - \frac{2}{2} or u = 0 or u \geq \frac{2}{2}

u \leq - \frac{2}{2} or u = 0 or u \geq \frac{2}{2}

u \leq - \frac{2}{2} or u = 0 or u \geq \frac{2}{2}

Substitute back

u = sin (2 x)

sin (2 x) \leq - \frac{2}{2} or sin (2 x) = 0 or sin (2 x) \geq \frac{2}{2}

sin (2 x) \leq - \frac{2}{2} : - \frac{3 π}{8} + πn \leq x \leq - \frac{π}{8} + πn

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

For

sin (x) \leq a

, if

- 1 < a < 1

then

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

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

a \leq u \leq b

then

a \leq u and u \leq b

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

- π - arcsin (- \frac{2}{2}) + 2 πn \leq 2 x : x \geq - \frac{3 π}{8} + πn

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

Switch sides

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

Simplify

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

- π - arcsin (- \frac{2}{2}) + 2 πn

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}) + 2 πn

Apply rule

- (- a) = a

= - π + \frac{π}{4} + 2 πn

2 x \geq - π + \frac{π}{4} + 2 πn

Divide both sides by

2

2 x \geq - π + \frac{π}{4} + 2 πn

Divide both sides by

2

\frac{2 x}{2} \geq - \frac{π}{2} + \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} \geq - \frac{π}{2} + \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- \frac{π}{2} + \frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : - \frac{π}{2} + \frac{π}{8} + πn

- \frac{π}{2} + \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= - \frac{π}{2} + \frac{π}{8} + πn

x \geq - \frac{π}{2} + \frac{π}{8} + πn

x \geq - \frac{π}{2} + \frac{π}{8} + πn

Simplify

- \frac{π}{2} + \frac{π}{8} : - \frac{3 π}{8}

- \frac{π}{2} + \frac{π}{8}

Least Common Multiplier of

2, 8 : 8

2, 8

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

8

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

8

For

\frac{π}{2} :

multiply the denominator and numerator by

4

\frac{π}{2} = \frac{π 4}{2 \cdot 4} = \frac{π 4}{8}

= - \frac{π 4}{8} + \frac{π}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 4 + π}{8}

Add similar elements:

- 4 π + π = - 3 π

= \frac{- 3 π}{8}

Apply the fraction rule:

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

= - \frac{3 π}{8}

x \geq - \frac{3 π}{8} + πn

x \geq - \frac{3 π}{8} + πn

2 x \leq arcsin (- \frac{2}{2}) + 2 πn : x \leq - \frac{π}{8} + πn

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

Simplify

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

arcsin (- \frac{2}{2}) + 2 πn

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}

= - \frac{π}{4} + 2 πn

2 x \leq - \frac{π}{4} + 2 πn

Divide both sides by

2

2 x \leq - \frac{π}{4} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- \frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : - \frac{π}{8} + πn

- \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= - \frac{π}{8} + πn

x \leq - \frac{π}{8} + πn

x \leq - \frac{π}{8} + πn

x \leq - \frac{π}{8} + πn

Combine the intervals

x \geq - \frac{3 π}{8} + πn and x \leq - \frac{π}{8} + πn

Merge Overlapping Intervals

- \frac{3 π}{8} + πn \leq x \leq - \frac{π}{8} + πn

sin (2 x) = 0 : x = πn, x = \frac{π}{2} + πn

sin (2 x) = 0

General solutions for

sin (2 x) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 x = 0 + 2 πn, 2 x = π + 2 πn

2 x = 0 + 2 πn, 2 x = π + 2 πn

Solve

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

Divide both sides by

2

2 x = 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplify

x = πn

x = πn

Solve

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Divide both sides by

2

2 x = π + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = πn, x = \frac{π}{2} + πn

sin (2 x) \geq \frac{2}{2} : \frac{π}{8} + πn \leq x \leq \frac{3 π}{8} + πn

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

a \leq u \leq b

then

a \leq u and u \leq b

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

arcsin (\frac{2}{2}) + 2 πn \leq 2 x : x \geq \frac{π}{8} + πn

arcsin (\frac{2}{2}) + 2 πn \leq 2 x

Switch sides

2 x \geq arcsin (\frac{2}{2}) + 2 πn

Simplify

arcsin (\frac{2}{2}) + 2 πn : \frac{π}{4} + 2 πn

arcsin (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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} + 2 πn

2 x \geq \frac{π}{4} + 2 πn

Divide both sides by

2

2 x \geq \frac{π}{4} + 2 πn

Divide both sides by

2

\frac{2 x}{2} \geq \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} \geq \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π}{8} + πn

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{8} + πn

x \geq \frac{π}{8} + πn

x \geq \frac{π}{8} + πn

x \geq \frac{π}{8} + πn

2 x \leq π - arcsin (\frac{2}{2}) + 2 πn : x \leq \frac{3 π}{8} + πn

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

Simplify

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

π - arcsin (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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}

= π - \frac{π}{4} + 2 πn

2 x \leq π - \frac{π}{4} + 2 πn

Divide both sides by

2

2 x \leq π - \frac{π}{4} + 2 πn

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{π}{2} - \frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π}{2} - \frac{π}{8} + πn

\frac{π}{2} - \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

= \frac{π}{2} - \frac{π}{8} + πn

x \leq \frac{π}{2} - \frac{π}{8} + πn

x \leq \frac{π}{2} - \frac{π}{8} + πn

Simplify

\frac{π}{2} - \frac{π}{8} : \frac{3 π}{8}

\frac{π}{2} - \frac{π}{8}

Least Common Multiplier of

2, 8 : 8

2, 8

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

8 : 2 \cdot 2 \cdot 2

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

2

8

= 2 \cdot 2 \cdot 2

Multiply the numbers:

2 \cdot 2 \cdot 2 = 8

= 8

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

8

For

\frac{π}{2} :

multiply the denominator and numerator by

4

\frac{π}{2} = \frac{π 4}{2 \cdot 4} = \frac{π 4}{8}

= \frac{π 4}{8} - \frac{π}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 - π}{8}

Add similar elements:

4 π - π = 3 π

= \frac{3 π}{8}

x \leq \frac{3 π}{8} + πn

x \leq \frac{3 π}{8} + πn

Combine the intervals

x \geq \frac{π}{8} + πn and x \leq \frac{3 π}{8} + πn

Merge Overlapping Intervals

\frac{π}{8} + πn \leq x \leq \frac{3 π}{8} + πn

Combine the intervals

- \frac{3 π}{8} + πn \leq x \leq - \frac{π}{8} + πn or x = \frac{π}{2} + πn or \frac{π}{8} + πn \leq x \leq \frac{3 π}{8} + πn

Merge Overlapping Intervals

\frac{π}{8} + πn \leq x \leq \frac{3 π}{8} + πn or x = \frac{π}{2} + πn or - \frac{3 π}{8} + πn \leq x \leq - \frac{π}{8} + πn

Popular Examples

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