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Popular Trigonometry >

sin((-pi)/6+3x)= 1/(sqrt(2))

Solution

sin (\frac{- π}{6} + 3 x) = \frac{1}{2}

Solution

x = \frac{2 πn}{3} + \frac{5 π}{36}, x = \frac{2 πn}{3} + \frac{11 π}{36}

Radians

x = \frac{5 π}{36} + \frac{2 π}{3} n, x = \frac{11 π}{36} + \frac{2 π}{3} n

Solution steps

sin (\frac{- π}{6} + 3 x) = \frac{1}{2}

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

sin (\frac{- π}{6} + 3 x) = \frac{2}{2}

General solutions for

sin (\frac{- π}{6} + 3 x) = \frac{2}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

Solve

\frac{- π}{6} + 3 x = \frac{π}{4} + 2 πn : x = \frac{2 πn}{3} + \frac{5 π}{36}

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

Move

\frac{- π}{6}

to the right side

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

Subtract

\frac{- π}{6}

from both sides

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

Simplify

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

\frac{π}{4} + 2 πn - \frac{- π}{6} : 2 πn + \frac{5 π}{12}

\frac{π}{4} + 2 πn - \frac{- π}{6}

Group like terms

= 2 πn + \frac{π}{4} - \frac{- π}{6}

Apply the fraction rule:

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

= 2 πn + \frac{π}{4} - (- \frac{π}{6})

Apply rule

- (- a) = a

= 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 = 5 π

= 2 πn + \frac{5 π}{12}

3 x = 2 πn + \frac{5 π}{12}

3 x = 2 πn + \frac{5 π}{12}

3 x = 2 πn + \frac{5 π}{12}

Divide both sides by

3

3 x = 2 πn + \frac{5 π}{12}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{5 π}{12}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{5 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{5 π}{12}}{3} : \frac{2 πn}{3} + \frac{5 π}{36}

\frac{2 πn}{3} + \frac{\frac{5 π}{12}}{3}

\frac{\frac{5 π}{12}}{3} = \frac{5 π}{36}

\frac{\frac{5 π}{12}}{3}

Apply the fraction rule:

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

= \frac{5 π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{5 π}{36}

= \frac{2 πn}{3} + \frac{5 π}{36}

x = \frac{2 πn}{3} + \frac{5 π}{36}

x = \frac{2 πn}{3} + \frac{5 π}{36}

x = \frac{2 πn}{3} + \frac{5 π}{36}

Solve

\frac{- π}{6} + 3 x = \frac{3 π}{4} + 2 πn : x = \frac{2 πn}{3} + \frac{11 π}{36}

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

Move

\frac{- π}{6}

to the right side

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

Subtract

\frac{- π}{6}

from both sides

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

Simplify

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

\frac{3 π}{4} + 2 πn - \frac{- π}{6} : 2 πn + \frac{11 π}{12}

\frac{3 π}{4} + 2 πn - \frac{- π}{6}

Group like terms

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

Apply the fraction rule:

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

= 2 πn + \frac{3 π}{4} - (- \frac{π}{6})

Apply rule

- (- a) = a

= 2 πn + \frac{3 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{3 π}{4} :

multiply the denominator and numerator by

3

\frac{3 π}{4} = \frac{3 π 3}{4 \cdot 3} = \frac{9 π}{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{9 π}{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{9 π + π 2}{12}

Add similar elements:

9 π + 2 π = 11 π

= 2 πn + \frac{11 π}{12}

3 x = 2 πn + \frac{11 π}{12}

3 x = 2 πn + \frac{11 π}{12}

3 x = 2 πn + \frac{11 π}{12}

Divide both sides by

3

3 x = 2 πn + \frac{11 π}{12}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{11 π}{12}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{11 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{11 π}{12}}{3} : \frac{2 πn}{3} + \frac{11 π}{36}

\frac{2 πn}{3} + \frac{\frac{11 π}{12}}{3}

\frac{\frac{11 π}{12}}{3} = \frac{11 π}{36}

\frac{\frac{11 π}{12}}{3}

Apply the fraction rule:

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

= \frac{11 π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{11 π}{36}

= \frac{2 πn}{3} + \frac{11 π}{36}

x = \frac{2 πn}{3} + \frac{11 π}{36}

x = \frac{2 πn}{3} + \frac{11 π}{36}

x = \frac{2 πn}{3} + \frac{11 π}{36}

x = \frac{2 πn}{3} + \frac{5 π}{36}, x = \frac{2 πn}{3} + \frac{11 π}{36}

Graph

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