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Popular >

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

Solution

sin (3 x - \frac{π}{3}) = \frac{1}{2}

Solution

x = \frac{2 πn}{3} + \frac{7 π}{36}, x = \frac{2 πn}{3} + \frac{13 π}{36}

Radians

x = \frac{7 π}{36} + \frac{2 π}{3} n, x = \frac{13 π}{36} + \frac{2 π}{3} n

Solution steps

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

General solutions for

sin (3 x - \frac{π}{3}) = \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}

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

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

Solve

3 x - \frac{π}{3} = \frac{π}{4} + 2 πn : x = \frac{2 πn}{3} + \frac{7 π}{36}

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

3 x - \frac{π}{3} + \frac{π}{3} : 3 x

3 x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 3 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}

3 x = 2 πn + \frac{7 π}{12}

3 x = 2 πn + \frac{7 π}{12}

3 x = 2 πn + \frac{7 π}{12}

Divide both sides by

3

3 x = 2 πn + \frac{7 π}{12}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{7 π}{12}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{7 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{7 π}{12}}{3} : \frac{2 πn}{3} + \frac{7 π}{36}

\frac{2 πn}{3} + \frac{\frac{7 π}{12}}{3}

\frac{\frac{7 π}{12}}{3} = \frac{7 π}{36}

\frac{\frac{7 π}{12}}{3}

Apply the fraction rule:

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

= \frac{7 π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{7 π}{36}

= \frac{2 πn}{3} + \frac{7 π}{36}

x = \frac{2 πn}{3} + \frac{7 π}{36}

x = \frac{2 πn}{3} + \frac{7 π}{36}

x = \frac{2 πn}{3} + \frac{7 π}{36}

Solve

3 x - \frac{π}{3} = \frac{3 π}{4} + 2 πn : x = \frac{2 πn}{3} + \frac{13 π}{36}

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

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

Simplify

3 x - \frac{π}{3} + \frac{π}{3} : 3 x

3 x - \frac{π}{3} + \frac{π}{3}

Add similar elements:

- \frac{π}{3} + \frac{π}{3} = 0

= 3 x

Simplify

\frac{3 π}{4} + 2 πn + \frac{π}{3} : 2 πn + \frac{13 π}{12}

\frac{3 π}{4} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{3 π}{4}

Least Common Multiplier of

3, 4 : 12

3, 4

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

3

4

= 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 2 \cdot 2 = 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}{3} :

multiply the denominator and numerator by

4

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

= \frac{π 4}{12} + \frac{9 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 + 9 π}{12}

Add similar elements:

4 π + 9 π = 13 π

= 2 πn + \frac{13 π}{12}

3 x = 2 πn + \frac{13 π}{12}

3 x = 2 πn + \frac{13 π}{12}

3 x = 2 πn + \frac{13 π}{12}

Divide both sides by

3

3 x = 2 πn + \frac{13 π}{12}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{13 π}{12}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{13 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{13 π}{12}}{3} : \frac{2 πn}{3} + \frac{13 π}{36}

\frac{2 πn}{3} + \frac{\frac{13 π}{12}}{3}

\frac{\frac{13 π}{12}}{3} = \frac{13 π}{36}

\frac{\frac{13 π}{12}}{3}

Apply the fraction rule:

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

= \frac{13 π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{13 π}{36}

= \frac{2 πn}{3} + \frac{13 π}{36}

x = \frac{2 πn}{3} + \frac{13 π}{36}

x = \frac{2 πn}{3} + \frac{13 π}{36}

x = \frac{2 πn}{3} + \frac{13 π}{36}

x = \frac{2 πn}{3} + \frac{7 π}{36}, x = \frac{2 πn}{3} + \frac{13 π}{36}

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