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Popular Trigonometry >

2sin^2(x/4)<1.5

Solution

2 sin^{2} (\frac{x}{4}) < 1.5

Solution

8 πn \leq x < \frac{4 π}{3} + 8 πn or \frac{8 π}{3} + 8 πn < x < \frac{16 π}{3} + 8 πn or \frac{20 π}{3} + 8 πn < x < 8 π + 8 πn

Interval Notation

[8 πn, \frac{4 π}{3} + 8 πn) \cup (\frac{8 π}{3} + 8 πn, \frac{16 π}{3} + 8 πn) \cup (\frac{20 π}{3} + 8 πn, 8 π + 8 πn)

Decimal

8 πn \leq x < 4.18879 \dots + 8 πn or 8.37758 \dots + 8 πn < x < 16.75516 \dots + 8 πn or 20.94395 \dots + 8 πn < x < 25.13274 \dots + 8 πn

Solution steps

2 sin^{2} (\frac{x}{4}) < 1.5

Divide both sides by

2

2 sin^{2} (\frac{x}{4}) < 1.5

Divide both sides by

2

\frac{2 sin ^{2} ( \frac{x}{4} )}{2} < \frac{1.5}{2}

Simplify

sin^{2} (\frac{x}{4}) < 0.75

sin^{2} (\frac{x}{4}) < 0.75

For

u^{n} < a

, if

n

is even then

- 0.75 < sin (\frac{x}{4}) < 0.75

0.75 = 0.86602 \dots

0.75

0.75 = 0.86602 \dots

= 0.86602 \dots

- 0.86602 \dots < sin (\frac{x}{4}) < 0.86602 \dots

a < u < b

then

a < u and u < b

- 0.86602 \dots < sin (\frac{x}{4}) and sin (\frac{x}{4}) < 0.86602 \dots

- 0.86602 \dots < sin (\frac{x}{4}) : - \frac{4 π}{3} + 8 πn < x < \frac{16 π}{3} + 8 πn

- 0.86602 \dots < sin (\frac{x}{4})

Switch sides

sin (\frac{x}{4}) > - 0.86602 \dots

For

sin (x) > a

, if

- 1 \leq a < 1

then

arcsin (a) + 2 πn < x < π - arcsin (a) + 2 πn

arcsin (- 0.86602 \dots) + 2 πn < \frac{x}{4} < π - arcsin (- 0.86602 \dots) + 2 πn

a < u < b

then

a < u and u < b

arcsin (- 0.86602 \dots) + 2 πn < \frac{x}{4} and \frac{x}{4} < π - arcsin (- 0.86602 \dots) + 2 πn

arcsin (- 0.86602 \dots) + 2 πn < \frac{x}{4} : x > - \frac{4 π}{3} + 8 πn

arcsin (- 0.86602 \dots) + 2 πn < \frac{x}{4}

Switch sides

\frac{x}{4} > arcsin (- 0.86602 \dots) + 2 πn

Simplify

arcsin (- 0.86602 \dots) + 2 πn : - \frac{π}{3} + 2 πn

arcsin (- 0.86602 \dots) + 2 πn

arcsin (- 0.86602 \dots) = - \frac{π}{3}

arcsin (- 0.86602 \dots)

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- 0.86602 \dots) = - arcsin (0.86602 \dots)

= - arcsin (0.86602 \dots)

Use the following trivial identity:

arcsin (0.86602 \dots) = \frac{π}{3}

arcsin (0.86602 \dots)

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}{3}

= - \frac{π}{3}

= - \frac{π}{3} + 2 πn

\frac{x}{4} > - \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{x}{4} > - \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{4 x}{4} > - 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} > - 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

- 4 \cdot \frac{π}{3} + 4 \cdot 2 πn : - \frac{4 π}{3} + 8 πn

- 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Multiply

4 \cdot \frac{π}{3} : \frac{4 π}{3}

4 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 4}{3}

= - \frac{4 π}{3} + 4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= - \frac{4 π}{3} + 8 πn

x > - \frac{4 π}{3} + 8 πn

x > - \frac{4 π}{3} + 8 πn

x > - \frac{4 π}{3} + 8 πn

\frac{x}{4} < π - arcsin (- 0.86602 \dots) + 2 πn : x < \frac{16 π}{3} + 8 πn

\frac{x}{4} < π - arcsin (- 0.86602 \dots) + 2 πn

Simplify

π - arcsin (- 0.86602 \dots) + 2 πn : π + \frac{π}{3} + 2 πn

π - arcsin (- 0.86602 \dots) + 2 πn

arcsin (- 0.86602 \dots) = - \frac{π}{3}

arcsin (- 0.86602 \dots)

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- 0.86602 \dots) = - arcsin (0.86602 \dots)

= - arcsin (0.86602 \dots)

Use the following trivial identity:

arcsin (0.86602 \dots) = \frac{π}{3}

arcsin (0.86602 \dots)

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}{3}

= - \frac{π}{3}

= π - (- \frac{π}{3}) + 2 πn

Apply rule

- (- a) = a

= π + \frac{π}{3} + 2 πn

\frac{x}{4} < π + \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{x}{4} < π + \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{4 x}{4} < 4 π + 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} < 4 π + 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

4 π + 4 \cdot \frac{π}{3} + 4 \cdot 2 πn : 4 π + \frac{4 π}{3} + 8 πn

4 π + 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Multiply

4 \cdot \frac{π}{3} : \frac{4 π}{3}

4 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 4}{3}

= 4 π + \frac{4 π}{3} + 4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 4 π + \frac{4 π}{3} + 8 πn

x < 4 π + \frac{4 π}{3} + 8 πn

x < 4 π + \frac{4 π}{3} + 8 πn

Simplify

4 π + \frac{4 π}{3} : \frac{16 π}{3}

4 π + \frac{4 π}{3}

Convert element to fraction:

4 π = \frac{4 π 3}{3}

= \frac{4 π 3}{3} + \frac{4 π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{4 π 3 + 4 π}{3}

4 π 3 + 4 π = 16 π

4 π 3 + 4 π

Multiply the numbers:

4 \cdot 3 = 12

= 12 π + 4 π

Add similar elements:

12 π + 4 π = 16 π

= 16 π

= \frac{16 π}{3}

x < \frac{16 π}{3} + 8 πn

x < \frac{16 π}{3} + 8 πn

Combine the intervals

x > - \frac{4 π}{3} + 8 πn and x < \frac{16 π}{3} + 8 πn

Merge Overlapping Intervals

- \frac{4 π}{3} + 8 πn < x < \frac{16 π}{3} + 8 πn

sin (\frac{x}{4}) < 0.86602 \dots : - \frac{16 π}{3} + 8 πn < x < \frac{4 π}{3} + 8 πn

sin (\frac{x}{4}) < 0.86602 \dots

For

sin (x) < a

, if

- 1 < a \leq 1

then

- π - arcsin (a) + 2 πn < x < arcsin (a) + 2 πn

- π - arcsin (0.86602 \dots) + 2 πn < \frac{x}{4} < arcsin (0.86602 \dots) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (0.86602 \dots) + 2 πn < \frac{x}{4} and \frac{x}{4} < arcsin (0.86602 \dots) + 2 πn

- π - arcsin (0.86602 \dots) + 2 πn < \frac{x}{4} : x > - \frac{16 π}{3} + 8 πn

- π - arcsin (0.86602 \dots) + 2 πn < \frac{x}{4}

Switch sides

\frac{x}{4} > - π - arcsin (0.86602 \dots) + 2 πn

Simplify

- π - arcsin (0.86602 \dots) + 2 πn : - π - \frac{π}{3} + 2 πn

- π - arcsin (0.86602 \dots) + 2 πn

Use the following trivial identity:

arcsin (0.86602 \dots) = \frac{π}{3}

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}{3} + 2 πn

\frac{x}{4} > - π - \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{x}{4} > - π - \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{4 x}{4} > - 4 π - 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} > - 4 π - 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

- 4 π - 4 \cdot \frac{π}{3} + 4 \cdot 2 πn : - 4 π - \frac{4 π}{3} + 8 πn

- 4 π - 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Multiply

4 \cdot \frac{π}{3} : \frac{4 π}{3}

4 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 4}{3}

= - 4 π - \frac{4 π}{3} + 4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= - 4 π - \frac{4 π}{3} + 8 πn

x > - 4 π - \frac{4 π}{3} + 8 πn

x > - 4 π - \frac{4 π}{3} + 8 πn

Simplify

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

- 4 π - \frac{4 π}{3}

Convert element to fraction:

4 π = \frac{4 π 3}{3}

= - \frac{4 π 3}{3} - \frac{4 π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{- 4 π 3 - 4 π}{3}

- 4 π 3 - 4 π = - 16 π

- 4 π 3 - 4 π

Multiply the numbers:

4 \cdot 3 = 12

= - 12 π - 4 π

Add similar elements:

- 12 π - 4 π = - 16 π

= - 16 π

= \frac{- 16 π}{3}

Apply the fraction rule:

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

= - \frac{16 π}{3}

x > - \frac{16 π}{3} + 8 πn

x > - \frac{16 π}{3} + 8 πn

\frac{x}{4} < arcsin (0.86602 \dots) + 2 πn : x < \frac{4 π}{3} + 8 πn

\frac{x}{4} < arcsin (0.86602 \dots) + 2 πn

Simplify

arcsin (0.86602 \dots) + 2 πn : \frac{π}{3} + 2 πn

arcsin (0.86602 \dots) + 2 πn

Use the following trivial identity:

arcsin (0.86602 \dots) = \frac{π}{3}

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}{3} + 2 πn

\frac{x}{4} < \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{x}{4} < \frac{π}{3} + 2 πn

Multiply both sides by

4

\frac{4 x}{4} < 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} < 4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

4 \cdot \frac{π}{3} + 4 \cdot 2 πn : \frac{4 π}{3} + 8 πn

4 \cdot \frac{π}{3} + 4 \cdot 2 πn

Multiply

4 \cdot \frac{π}{3} : \frac{4 π}{3}

4 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 4}{3}

= \frac{4 π}{3} + 4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= \frac{4 π}{3} + 8 πn

x < \frac{4 π}{3} + 8 πn

x < \frac{4 π}{3} + 8 πn

x < \frac{4 π}{3} + 8 πn

Combine the intervals

x > - \frac{16 π}{3} + 8 πn and x < \frac{4 π}{3} + 8 πn

Merge Overlapping Intervals

- \frac{16 π}{3} + 8 πn < x < \frac{4 π}{3} + 8 πn

Combine the intervals

- \frac{4 π}{3} + 8 πn < x < \frac{16 π}{3} + 8 πn and - \frac{16 π}{3} + 8 πn < x < \frac{4 π}{3} + 8 πn

Merge Overlapping Intervals

8 πn \leq x < \frac{4 π}{3} + 8 πn or \frac{8 π}{3} + 8 πn < x < \frac{16 π}{3} + 8 πn or \frac{20 π}{3} + 8 πn < x < 8 π + 8 πn

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