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sin(x)<= 1/2 ,0<= x<= 2pi

Solution

sin (x) \leq \frac{1}{2}, 0 \leq x \leq 2 π

0 \leq x \leq \frac{π}{6} or 2 π - \frac{7 π}{6} \leq x \leq 2 π

Interval Notation

[0, \frac{π}{6}] \cup [2 π - \frac{7 π}{6}, 2 π]

Decimal Notation

0 \leq x \leq 0.52359 \dots or 2.61799 \dots \leq x \leq 6.28318 \dots

Solution steps

sin (x) \leq \frac{1}{2}, 0 \leq x \leq 2 π

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{1}{2}) + 2 πn \leq x \leq arcsin (\frac{1}{2}) + 2 πn

Simplify

- π - arcsin (\frac{1}{2}) : - \frac{7 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 = - 7 π

= \frac{- 7 π}{6}

Apply the fraction rule:

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

= - \frac{7 π}{6}

= - \frac{7 π}{6}

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}

- \frac{7 π}{6} + 2 πn \leq x \leq \frac{π}{6} + 2 πn

Combine the intervals

- \frac{7 π}{6} + 2 πn \leq x \leq \frac{π}{6} + 2 πn and 0 \leq x \leq 2 π

0 \leq x \leq \frac{π}{6} or 2 π - \frac{7 π}{6} \leq x \leq 2 π

cos (x - y) \geq 0

cos (x) \geq \frac{- 1}{2}

\frac{cos ^{2} ( x )}{1 - cos ( x )} < 0

\frac{π}{2} - arctan (n) \leq 1 0^{- 3}

cos (3 x) \leq \frac{3}{2}