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Popular >

sin(x)>sqrt(3)cos(x)

Solution

sin (x) > 3 cos (x)

Solution

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

Interval Notation

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

Decimal Notation

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

Solution steps

sin (x) > 3 cos (x)

Move

3 cos (x)

to the left side

sin (x) > 3 cos (x)

Subtract

3 cos (x)

from both sides

sin (x) - 3 cos (x) > 3 cos (x) - 3 cos (x)

sin (x) - 3 cos (x) > 0

sin (x) - 3 cos (x) > 0

Rewrite using trig identities

Divide both sides by

2

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

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

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

Expand

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

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

Apply the fraction rule:

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

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

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

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

Expand

\frac{0}{2} : 0

\frac{0}{2}

Apply rule

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

= 0

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

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

\frac{3}{2} = sin (\frac{π}{3})

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

\frac{1}{2} = cos (\frac{π}{3})

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

Use the following identity:

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

sin (x - \frac{π}{3}) > 0

sin (x - \frac{π}{3}) > 0

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

a < u < b

then

a < u and u < b

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

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

arcsin (0) + 2 πn < x - \frac{π}{3}

Switch sides

x - \frac{π}{3} > 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

x - \frac{π}{3} > 2 πn

Move

\frac{π}{3}

to the right side

x - \frac{π}{3} > 2 πn

Add

\frac{π}{3}

to both sides

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

Simplify

x > 2 πn + \frac{π}{3}

x > 2 πn + \frac{π}{3}

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

x - \frac{π}{3} < 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

x - \frac{π}{3} < π + 2 πn

Move

\frac{π}{3}

to the right side

x - \frac{π}{3} < π + 2 πn

Add

\frac{π}{3}

to both sides

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

Simplify

x < π + 2 πn + \frac{π}{3}

x < 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 < \frac{4 π}{3} + 2 πn

Combine the intervals

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

Merge Overlapping Intervals

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

Popular Examples

-cos(x)>0

- cos (x) > 0

tan(x)>= 3

tan (x) \geq 3

2cos^2(x)< 3/2

2 cos^{2} (x) < \frac{3}{2}

sin(3x)cos(3x)-1/4 >= 0

sin (3 x) cos (3 x) - \frac{1}{4} \geq 0

sin(x)+sqrt(3)cos(x)>0

sin (x) + 3 cos (x) > 0