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Popular Trigonometry >

sin(3x-pi/6)+cos(3x-pi/6)>0

Solution

sin (3 x - \frac{π}{6}) + cos (3 x - \frac{π}{6}) > 0

Solution

- \frac{π}{36} + \frac{2 π}{3} n < x < \frac{11 π}{36} + \frac{2 π}{3} n

Interval Notation

(- \frac{π}{36} + \frac{2 π}{3} n, \frac{11 π}{36} + \frac{2 π}{3} n)

Decimal

- 0.08726 \dots + \frac{2 π}{3} n < x < 0.95993 \dots + \frac{2 π}{3} n

Solution steps

sin (3 x - \frac{π}{6}) + cos (3 x - \frac{π}{6}) > 0

Use the following identity:

cos (x) + sin (x) = 2 sin (\frac{π}{4} + x)

2 sin (\frac{π}{4} - \frac{π}{6} + 3 x) > 0

Divide both sides by

2

2 sin (\frac{π}{4} - \frac{π}{6} + 3 x) > 0

Divide both sides by

2

\frac{2 sin ( \frac{π}{4} - \frac{π}{6} + 3 x )}{2} > \frac{0}{2}

Simplify

sin (\frac{π}{4} - \frac{π}{6} + 3 x) > 0

sin (\frac{π}{4} - \frac{π}{6} + 3 x) > 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 < (\frac{π}{4} - \frac{π}{6} + 3 x) < π - arcsin (0) + 2 πn

a < u < b

then

a < u and u < b

arcsin (0) + 2 πn < \frac{π}{4} - \frac{π}{6} + 3 x and \frac{π}{4} - \frac{π}{6} + 3 x < π - arcsin (0) + 2 πn

arcsin (0) + 2 πn < \frac{π}{4} - \frac{π}{6} + 3 x : x > \frac{2 πn}{3} - \frac{π}{36}

arcsin (0) + 2 πn < \frac{π}{4} - \frac{π}{6} + 3 x

Switch sides

\frac{π}{4} - \frac{π}{6} + 3 x > 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}{4} - \frac{π}{6} + 3 x > 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} - \frac{π}{6} + 3 x > 2 πn

Subtract

\frac{π}{4}

from both sides

\frac{π}{4} - \frac{π}{6} + 3 x - \frac{π}{4} > 2 πn - \frac{π}{4}

Simplify

- \frac{π}{6} + 3 x > 2 πn - \frac{π}{4}

- \frac{π}{6} + 3 x > 2 πn - \frac{π}{4}

Move

\frac{π}{6}

to the right side

- \frac{π}{6} + 3 x > 2 πn - \frac{π}{4}

Add

\frac{π}{6}

to both sides

- \frac{π}{6} + 3 x + \frac{π}{6} > 2 πn - \frac{π}{4} + \frac{π}{6}

Simplify

- \frac{π}{6} + 3 x + \frac{π}{6} > 2 πn - \frac{π}{4} + \frac{π}{6}

Simplify

- \frac{π}{6} + 3 x + \frac{π}{6} : 3 x

- \frac{π}{6} + 3 x + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} > 0

= 3 x

Simplify

2 πn - \frac{π}{4} + \frac{π}{6} : 2 πn - \frac{π}{12}

2 πn - \frac{π}{4} + \frac{π}{6}

Least Common Multiplier of

4, 6 : 12

4, 6

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

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

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

4

6

= 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}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

= - \frac{π 3}{12} + \frac{π 2}{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 2}{12}

Add similar elements:

- 3 π + 2 π = - π

= \frac{- π}{12}

Apply the fraction rule:

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

= 2 πn - \frac{π}{12}

3 x > 2 πn - \frac{π}{12}

3 x > 2 πn - \frac{π}{12}

3 x > 2 πn - \frac{π}{12}

Divide both sides by

3

3 x > 2 πn - \frac{π}{12}

Divide both sides by

3

\frac{3 x}{3} > \frac{2 πn}{3} - \frac{\frac{π}{12}}{3}

Simplify

\frac{3 x}{3} > \frac{2 πn}{3} - \frac{\frac{π}{12}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} - \frac{\frac{π}{12}}{3} : \frac{2 πn}{3} - \frac{π}{36}

\frac{2 πn}{3} - \frac{\frac{π}{12}}{3}

\frac{\frac{π}{12}}{3} = \frac{π}{36}

\frac{\frac{π}{12}}{3}

Apply the fraction rule:

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

= \frac{π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{π}{36}

= \frac{2 πn}{3} - \frac{π}{36}

x > \frac{2 πn}{3} - \frac{π}{36}

x > \frac{2 πn}{3} - \frac{π}{36}

x > \frac{2 πn}{3} - \frac{π}{36}

\frac{π}{4} - \frac{π}{6} + 3 x < π - arcsin (0) + 2 πn : x < \frac{11 π}{36} + \frac{2 π}{3} n

\frac{π}{4} - \frac{π}{6} + 3 x < 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}{4} - \frac{π}{6} + 3 x < π + 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} - \frac{π}{6} + 3 x < π + 2 πn

Subtract

\frac{π}{4}

from both sides

\frac{π}{4} - \frac{π}{6} + 3 x - \frac{π}{4} < π + 2 πn - \frac{π}{4}

Simplify

- \frac{π}{6} + 3 x < π + 2 πn - \frac{π}{4}

- \frac{π}{6} + 3 x < π + 2 πn - \frac{π}{4}

Move

\frac{π}{6}

to the right side

- \frac{π}{6} + 3 x < π + 2 πn - \frac{π}{4}

Add

\frac{π}{6}

to both sides

- \frac{π}{6} + 3 x + \frac{π}{6} < π + 2 πn - \frac{π}{4} + \frac{π}{6}

Simplify

- \frac{π}{6} + 3 x + \frac{π}{6} < π + 2 πn - \frac{π}{4} + \frac{π}{6}

Simplify

- \frac{π}{6} + 3 x + \frac{π}{6} : 3 x

- \frac{π}{6} + 3 x + \frac{π}{6}

Add similar elements:

- \frac{π}{6} + \frac{π}{6} < 0

= 3 x

Simplify

π + 2 πn - \frac{π}{4} + \frac{π}{6} : π + 2 πn - \frac{π}{12}

π + 2 πn - \frac{π}{4} + \frac{π}{6}

Least Common Multiplier of

4, 6 : 12

4, 6

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

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

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

4

6

= 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}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

= - \frac{π 3}{12} + \frac{π 2}{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 2}{12}

Add similar elements:

- 3 π + 2 π = - π

= \frac{- π}{12}

Apply the fraction rule:

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

= π + 2 πn - \frac{π}{12}

3 x < π + 2 πn - \frac{π}{12}

3 x < π + 2 πn - \frac{π}{12}

3 x < π + 2 πn - \frac{π}{12}

Divide both sides by

3

3 x < π + 2 πn - \frac{π}{12}

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{π}{3} + \frac{2 πn}{3} - \frac{\frac{π}{12}}{3} : \frac{π}{3} - \frac{π}{36} + \frac{2 πn}{3}

\frac{π}{3} + \frac{2 πn}{3} - \frac{\frac{π}{12}}{3}

\frac{\frac{π}{12}}{3} = \frac{π}{36}

\frac{\frac{π}{12}}{3}

Apply the fraction rule:

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

= \frac{π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{π}{36}

= \frac{π}{3} + \frac{2 πn}{3} - \frac{π}{36}

Group like terms

= \frac{π}{3} - \frac{π}{36} + \frac{2 πn}{3}

x < \frac{π}{3} - \frac{π}{36} + \frac{2 πn}{3}

x < \frac{π}{3} - \frac{π}{36} + \frac{2 πn}{3}

Simplify

\frac{π}{3} - \frac{π}{36} : \frac{11 π}{36}

\frac{π}{3} - \frac{π}{36}

Least Common Multiplier of

3, 36 : 36

3, 36

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

36 : 2 \cdot 2 \cdot 3 \cdot 3

36

36

divides by

236 = 18 \cdot 2

= 2 \cdot 18

18

divides by

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 9

9

divides by

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 3

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

3

36

= 3 \cdot 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 3 \cdot 2 \cdot 2 = 36

= 36

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

36

For

\frac{π}{3} :

multiply the denominator and numerator by

12

\frac{π}{3} = \frac{π 12}{3 \cdot 12} = \frac{π 12}{36}

= \frac{π 12}{36} - \frac{π}{36}

Since the denominators are equal, combine the fractions:

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

= \frac{π 12 - π}{36}

Add similar elements:

12 π - π = 11 π

= \frac{11 π}{36}

x < \frac{11 π}{36} + \frac{2 π}{3} n

x < \frac{11 π}{36} + \frac{2 π}{3} n

Combine the intervals

x > \frac{2 πn}{3} - \frac{π}{36} and x < \frac{11 π}{36} + \frac{2 π}{3} n

Merge Overlapping Intervals

- \frac{π}{36} + \frac{2 π}{3} n < x < \frac{11 π}{36} + \frac{2 π}{3} n

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