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Popular Trigonometry >

1/2 <sin(θ)<(sqrt(3))/2

Solution

\frac{1}{2} < sin (θ) < \frac{3}{2}

Solution

\frac{π}{6} + 2 πn < θ < \frac{π}{3} + 2 πn or \frac{2 π}{3} + 2 πn < θ < \frac{5 π}{6} + 2 πn

Interval Notation

(\frac{π}{6} + 2 πn, \frac{π}{3} + 2 πn) \cup (\frac{2 π}{3} + 2 πn, \frac{5 π}{6} + 2 πn)

Decimal

0.52359 \dots + 2 πn < θ < 1.04719 \dots + 2 πn or 2.09439 \dots + 2 πn < θ < 2.61799 \dots + 2 πn

Solution steps

\frac{1}{2} < sin (θ) < \frac{3}{2}

a < u < b

then

a < u and u < b

\frac{1}{2} < sin (θ) and sin (θ) < \frac{3}{2}

\frac{1}{2} < sin (θ) : \frac{π}{6} + 2 πn < θ < \frac{5 π}{6} + 2 πn

\frac{1}{2} < sin (θ)

Switch sides

sin (θ) > \frac{1}{2}

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (\frac{1}{2}) + 2 πn < θ < π - arcsin (\frac{1}{2}) + 2 πn

Simplify

arcsin (\frac{1}{2}) : \frac{π}{6}

arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

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}{6}

Simplify

π - arcsin (\frac{1}{2}) : \frac{5 π}{6}

π - arcsin (\frac{1}{2})

Use the following trivial identity:

arcsin (\frac{1}{2}) = \frac{π}{6}

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}{6}

Simplify

π - \frac{π}{6}

Convert element to fraction:

π = \frac{π 6}{6}

= \frac{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{π 6 - π}{6}

Add similar elements:

6 π - π = 5 π

= \frac{5 π}{6}

= \frac{5 π}{6}

\frac{π}{6} + 2 πn < θ < \frac{5 π}{6} + 2 πn

sin (θ) < \frac{3}{2} : - \frac{4 π}{3} + 2 πn < θ < \frac{π}{3} + 2 πn

sin (θ) < \frac{3}{2}

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (\frac{3}{2}) + 2 πn < θ < arcsin (\frac{3}{2}) + 2 πn

Simplify

- π - arcsin (\frac{3}{2}) : - \frac{4 π}{3}

- π - arcsin (\frac{3}{2})

Use the following trivial identity:

arcsin (\frac{3}{2}) = \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}

Simplify

- π - \frac{π}{3}

Convert element to fraction:

π = \frac{π 3}{3}

= - \frac{π 3}{3} - \frac{π}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 - π}{3}

Add similar elements:

- 3 π - π = - 4 π

= \frac{- 4 π}{3}

Apply the fraction rule:

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

= - \frac{4 π}{3}

= - \frac{4 π}{3}

Simplify

arcsin (\frac{3}{2}) : \frac{π}{3}

arcsin (\frac{3}{2})

Use the following trivial identity:

arcsin (\frac{3}{2}) = \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}

- \frac{4 π}{3} + 2 πn < θ < \frac{π}{3} + 2 πn

Combine the intervals

\frac{π}{6} + 2 πn < θ < \frac{5 π}{6} + 2 πn and - \frac{4 π}{3} + 2 πn < θ < \frac{π}{3} + 2 πn

Merge Overlapping Intervals

\frac{π}{6} + 2 πn < θ < \frac{π}{3} + 2 πn or \frac{2 π}{3} + 2 πn < θ < \frac{5 π}{6} + 2 πn

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