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Popular Trigonometry >

sin(x)-sqrt(3)cos(x)<1

Solution

sin (x) - 3 cos (x) < 1

Solution

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

Interval Notation

(- \frac{5 π}{6} + 2 πn, \frac{π}{2} + 2 πn)

Decimal

- 2.61799 \dots + 2 πn < x < 1.57079 \dots + 2 πn

Solution steps

sin (x) - 3 cos (x) < 1

Rewrite using trig identities

Divide both sides by

2

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

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

\frac{3}{2} = sin (\frac{π}{3})

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

\frac{1}{2} = cos (\frac{π}{3})

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

Use the following identity:

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

sin (x - \frac{π}{3}) < \frac{1}{2}

sin (x - \frac{π}{3}) < \frac{1}{2}

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

- π - arcsin (\frac{1}{2}) + 2 πn < (x - \frac{π}{3}) < arcsin (\frac{1}{2}) + 2 πn

a < u < b

then

a < u and u < b

- π - arcsin (\frac{1}{2}) + 2 πn < x - \frac{π}{3} and x - \frac{π}{3} < arcsin (\frac{1}{2}) + 2 πn

- π - arcsin (\frac{1}{2}) + 2 πn < x - \frac{π}{3} : x > - \frac{5 π}{6} + 2 πn

- π - arcsin (\frac{1}{2}) + 2 πn < x - \frac{π}{3}

Switch sides

x - \frac{π}{3} > - π - arcsin (\frac{1}{2}) + 2 πn

Simplify

- π - arcsin (\frac{1}{2}) + 2 πn : - π - \frac{π}{6} + 2 πn

- π - arcsin (\frac{1}{2}) + 2 πn

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} + 2 πn

x - \frac{π}{3} > - π - \frac{π}{6} + 2 πn

Move

\frac{π}{3}

to the right side

x - \frac{π}{3} > - π - \frac{π}{6} + 2 πn

Add

\frac{π}{3}

to both sides

x - \frac{π}{3} + \frac{π}{3} > - π - \frac{π}{6} + 2 πn + \frac{π}{3}

Simplify

x - \frac{π}{3} + \frac{π}{3} > - π - \frac{π}{6} + 2 πn + \frac{π}{3}

Simplify

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} > 0

= x

Simplify

- π - \frac{π}{6} + 2 πn + \frac{π}{3} : - π + 2 πn + \frac{π}{6}

- π - \frac{π}{6} + 2 πn + \frac{π}{3}

Group like terms

= - π + 2 πn - \frac{π}{6} + \frac{π}{3}

Least Common Multiplier of

6, 3 : 6

6, 3

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

6

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= - \frac{π}{6} + \frac{π 2}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π + π 2}{6}

Add similar elements:

- π + 2 π = π

= - π + 2 πn + \frac{π}{6}

x > - π + 2 πn + \frac{π}{6}

x > - π + 2 πn + \frac{π}{6}

x > - 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}

Apply the fraction rule:

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

= - \frac{5 π}{6}

x > - \frac{5 π}{6} + 2 πn

x - \frac{π}{3} < arcsin (\frac{1}{2}) + 2 πn : x < 2 πn + \frac{π}{2}

x - \frac{π}{3} < arcsin (\frac{1}{2}) + 2 πn

Simplify

arcsin (\frac{1}{2}) + 2 πn : \frac{π}{6} + 2 πn

arcsin (\frac{1}{2}) + 2 πn

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} + 2 πn

x - \frac{π}{3} < \frac{π}{6} + 2 πn

Move

\frac{π}{3}

to the right side

x - \frac{π}{3} < \frac{π}{6} + 2 πn

Add

\frac{π}{3}

to both sides

x - \frac{π}{3} + \frac{π}{3} < \frac{π}{6} + 2 πn + \frac{π}{3}

Simplify

x - \frac{π}{3} + \frac{π}{3} < \frac{π}{6} + 2 πn + \frac{π}{3}

Simplify

x - \frac{π}{3} + \frac{π}{3} : x

x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} < 0

= x

Simplify

\frac{π}{6} + 2 πn + \frac{π}{3} : 2 πn + \frac{π}{2}

\frac{π}{6} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{6} + \frac{π}{3}

Least Common Multiplier of

6, 3 : 6

6, 3

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

6

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= \frac{π}{6} + \frac{π 2}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π + π 2}{6}

Add similar elements:

π + 2 π = 3 π

= \frac{3 π}{6}

Cancel the common factor:

3

= 2 πn + \frac{π}{2}

x < 2 πn + \frac{π}{2}

x < 2 πn + \frac{π}{2}

x < 2 πn + \frac{π}{2}

Combine the intervals

x > - \frac{5 π}{6} + 2 πn and x < 2 πn + \frac{π}{2}

Merge Overlapping Intervals

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

Popular Examples

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sin((pix)/2)> 1/2

sin(x)> 1/(sin(x))

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2sin(x)cos(x)>= (sqrt(3))/2