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Popular >

-(sqrt(3))/2 sin(x)-1/2 cos(x)<= 0

Solution

- \frac{3}{2} sin (x) - \frac{1}{2} cos (x) \leq 0

Solution

- \frac{π}{6} + 2 πn \leq x \leq \frac{5 π}{6} + 2 πn

Interval Notation

[- \frac{π}{6} + 2 πn, \frac{5 π}{6} + 2 πn]

Decimal Notation

- 0.52359 \dots + 2 πn \leq x \leq 2.61799 \dots + 2 πn

Solution steps

- \frac{3}{2} sin (x) - \frac{1}{2} cos (x) \leq 0

Rewrite using trig identities

Multiply both sides by -1 (reverse the inequality)

- \frac{3}{2} sin (x) (- 1) - \frac{1}{2} cos (x) (- 1) \geq 0 \cdot (- 1)

\frac{3}{2} sin (x) + \frac{1}{2} cos (x) \geq 0

\frac{3}{2} = cos (\frac{π}{6})

cos (\frac{π}{6}) sin (x) + \frac{1}{2} cos (x) \geq 0

\frac{1}{2} = sin (\frac{π}{6})

cos (\frac{π}{6}) sin (x) + sin (\frac{π}{6}) cos (x) \geq 0

Use the following identity:

cos (s) sin (t) + cos (t) sin (s) = sin (s + t)

sin (\frac{π}{6} + x) \geq 0

sin (\frac{π}{6} + x) \geq 0

For

sin (x) \geq a

, if

- 1 < a < 1

then

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

arcsin (0) + 2 πn \leq (\frac{π}{6} + x) \leq π - arcsin (0) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

arcsin (0) + 2 πn \leq \frac{π}{6} + x and \frac{π}{6} + x \leq π - arcsin (0) + 2 πn

arcsin (0) + 2 πn \leq \frac{π}{6} + x : x \geq 2 πn - \frac{π}{6}

arcsin (0) + 2 πn \leq \frac{π}{6} + x

Switch sides

\frac{π}{6} + x \geq arcsin (0) + 2 πn

Simplify

arcsin (0) + 2 πn : 2 πn

arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

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}

= 0 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

\frac{π}{6} + x \geq 2 πn

Move

\frac{π}{6}

to the right side

\frac{π}{6} + x \geq 2 πn

Subtract

\frac{π}{6}

from both sides

\frac{π}{6} + x - \frac{π}{6} \geq 2 πn - \frac{π}{6}

Simplify

x \geq 2 πn - \frac{π}{6}

x \geq 2 πn - \frac{π}{6}

\frac{π}{6} + x \leq π - arcsin (0) + 2 πn : x \leq \frac{5 π}{6} + 2 πn

\frac{π}{6} + x \leq π - arcsin (0) + 2 πn

Simplify

π - arcsin (0) + 2 πn : π + 2 πn

π - arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

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 - 0 + 2 πn

π - 0 + 2 πn = π + 2 πn

= π + 2 πn

\frac{π}{6} + x \leq π + 2 πn

Move

\frac{π}{6}

to the right side

\frac{π}{6} + x \leq π + 2 πn

Subtract

\frac{π}{6}

from both sides

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

Simplify

x \leq π + 2 πn - \frac{π}{6}

x \leq π + 2 πn - \frac{π}{6}

Simplify

π - \frac{π}{6} : \frac{5 π}{6}

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}

x \leq \frac{5 π}{6} + 2 πn

Combine the intervals

x \geq 2 πn - \frac{π}{6} and x \leq \frac{5 π}{6} + 2 πn

Merge Overlapping Intervals

- \frac{π}{6} + 2 πn \leq x \leq \frac{5 π}{6} + 2 πn

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