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Popular Trigonometry >

sqrt(3)cos(x)-sin(x)<= 0

Solution

3 cos (x) - sin (x) \leq 0

Solution

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

Interval Notation

[\frac{π}{3} + 2 πn, \frac{4 π}{3} + 2 πn]

Decimal

1.04719 \dots + 2 πn \leq x \leq 4.18879 \dots + 2 πn

Solution steps

3 cos (x) - sin (x) \leq 0

Rewrite using trig identities

Divide both sides by

2

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

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

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

Expand

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

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

Apply the fraction rule:

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

\frac{3 cos ( x ) - sin ( x )}{2} = \frac{3 cos ( x )}{2} - \frac{sin ( x )}{2}

= \frac{3 cos ( x )}{2} - \frac{sin ( x )}{2}

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

Expand

\frac{0}{2} : 0

\frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

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

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

\frac{3}{2} = sin (\frac{π}{3})

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

\frac{1}{2} = cos (\frac{π}{3})

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

Use the following identity:

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

sin (\frac{π}{3} - x) \leq 0

Factor out

- 1

from

\frac{π}{3} - x : - (- \frac{π}{3} + x)

sin (- (- \frac{π}{3} + x)) \leq 0

Use the following identity:

sin (- x) = - sin (x)

- sin (- \frac{π}{3} + x) \leq 0

Multiply both sides by

- 1

- sin (- \frac{π}{3} + x) \leq 0

Multiply both sides by -1 (reverse the inequality)

(- sin (- \frac{π}{3} + x)) (- 1) \geq 0 \cdot (- 1)

Simplify

sin (- \frac{π}{3} + x) \geq 0

sin (- \frac{π}{3} + x) \geq 0

sin (- \frac{π}{3} + 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}{3} + 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}{3} + x and - \frac{π}{3} + x \leq π - arcsin (0) + 2 πn

arcsin (0) + 2 πn \leq - \frac{π}{3} + x : x \geq 2 πn + \frac{π}{3}

arcsin (0) + 2 πn \leq - \frac{π}{3} + x

Switch sides

- \frac{π}{3} + 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}{3} + x \geq 2 πn

Move

\frac{π}{3}

to the right side

- \frac{π}{3} + x \geq 2 πn

Add

\frac{π}{3}

to both sides

- \frac{π}{3} + x + \frac{π}{3} \geq 2 πn + \frac{π}{3}

Simplify

x \geq 2 πn + \frac{π}{3}

x \geq 2 πn + \frac{π}{3}

- \frac{π}{3} + x \leq π - arcsin (0) + 2 πn : x \leq \frac{4 π}{3} + 2 πn

- \frac{π}{3} + 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}{3} + x \leq π + 2 πn

Move

\frac{π}{3}

to the right side

- \frac{π}{3} + x \leq π + 2 πn

Add

\frac{π}{3}

to both sides

- \frac{π}{3} + x + \frac{π}{3} \leq π + 2 πn + \frac{π}{3}

Simplify

x \leq π + 2 πn + \frac{π}{3}

x \leq π + 2 πn + \frac{π}{3}

Simplify

π + \frac{π}{3} : \frac{4 π}{3}

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}

x \leq \frac{4 π}{3} + 2 πn

Combine the intervals

x \geq 2 πn + \frac{π}{3} and x \leq \frac{4 π}{3} + 2 πn

Merge Overlapping Intervals

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

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