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Popular >

sin(x)> 1/(sqrt(2))

Solution

sin (x) > \frac{1}{2}

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

Interval Notation

(\frac{π}{4} + 2 πn, \frac{3 π}{4} + 2 πn)

Decimal Notation

0.78539 \dots + 2 πn < x < 2.35619 \dots + 2 πn

Solution steps

sin (x) > \frac{1}{2}

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}

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

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

Simplify

arcsin (\frac{2}{2}) : \frac{π}{4}

arcsin (\frac{2}{2})

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}

Simplify

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

π - arcsin (\frac{2}{2})

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}

Simplify

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 = 3 π

= \frac{3 π}{4}

= \frac{3 π}{4}

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

sin (θ) \leq \frac{1}{2}

2 - sin^{2} (x) \geq 0

cos (θ) < 0, tan (θ) > 0

3 cos^{2} (x) + 5 cos (x) - 2 < 0

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