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Popular Trigonometry >

sin(x)-sqrt(3)cos(x)>sqrt(2)

Solution

sin (x) - 3 cos (x) > 2

Solution

\frac{7 π}{12} + 2 πn < x < \frac{13 π}{12} + 2 πn

Interval Notation

(\frac{7 π}{12} + 2 πn, \frac{13 π}{12} + 2 πn)

Decimal

1.83259 \dots + 2 πn < x < 3.40339 \dots + 2 πn

Solution steps

sin (x) - 3 cos (x) > 2

Rewrite using trig identities

Divide both sides by

2

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

\frac{3}{2} = sin (\frac{π}{3})

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

\frac{1}{2} = cos (\frac{π}{3})

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

Use the following identity:

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

sin (x - \frac{π}{3}) > \frac{2}{2}

sin (x - \frac{π}{3}) > \frac{2}{2}

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

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

a < u < b

then

a < u and u < b

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

arcsin (\frac{2}{2}) + 2 πn < x - \frac{π}{3} : x > 2 πn + \frac{7 π}{12}

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

Switch sides

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

Simplify

arcsin (\frac{2}{2}) + 2 πn : \frac{π}{4} + 2 πn

arcsin (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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}{4} + 2 πn

x - \frac{π}{3} > \frac{π}{4} + 2 πn

Move

\frac{π}{3}

to the right side

x - \frac{π}{3} > \frac{π}{4} + 2 πn

Add

\frac{π}{3}

to both sides

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

Simplify

x - \frac{π}{3} + \frac{π}{3} > \frac{π}{4} + 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}{4} + 2 πn + \frac{π}{3} : 2 πn + \frac{7 π}{12}

\frac{π}{4} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{4} + \frac{π}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

4

3

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{π}{3} :

multiply the denominator and numerator by

4

\frac{π}{3} = \frac{π 4}{3 \cdot 4} = \frac{π 4}{12}

= \frac{π 3}{12} + \frac{π 4}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + π 4}{12}

Add similar elements:

3 π + 4 π = 7 π

= 2 πn + \frac{7 π}{12}

x > 2 πn + \frac{7 π}{12}

x > 2 πn + \frac{7 π}{12}

x > 2 πn + \frac{7 π}{12}

x - \frac{π}{3} < π - arcsin (\frac{2}{2}) + 2 πn : x < \frac{13 π}{12} + 2 πn

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

Simplify

π - arcsin (\frac{2}{2}) + 2 πn : π - \frac{π}{4} + 2 πn

π - arcsin (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

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}{4} + 2 πn

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

x - \frac{π}{3} + \frac{π}{3} < π - \frac{π}{4} + 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}{4} + 2 πn + \frac{π}{3} : π + 2 πn + \frac{π}{12}

π - \frac{π}{4} + 2 πn + \frac{π}{3}

Group like terms

= π + 2 πn - \frac{π}{4} + \frac{π}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

4

3

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{π}{3} :

multiply the denominator and numerator by

4

\frac{π}{3} = \frac{π 4}{3 \cdot 4} = \frac{π 4}{12}

= - \frac{π 3}{12} + \frac{π 4}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + π 4}{12}

Add similar elements:

- 3 π + 4 π = π

= π + 2 πn + \frac{π}{12}

x < π + 2 πn + \frac{π}{12}

x < π + 2 πn + \frac{π}{12}

x < π + 2 πn + \frac{π}{12}

Simplify

π + \frac{π}{12} : \frac{13 π}{12}

π + \frac{π}{12}

Convert element to fraction:

π = \frac{π 12}{12}

= \frac{π 12}{12} + \frac{π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 12 + π}{12}

Add similar elements:

12 π + π = 13 π

= \frac{13 π}{12}

x < \frac{13 π}{12} + 2 πn

Combine the intervals

x > 2 πn + \frac{7 π}{12} and x < \frac{13 π}{12} + 2 πn

Merge Overlapping Intervals

\frac{7 π}{12} + 2 πn < x < \frac{13 π}{12} + 2 πn

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