What is the general solution for sin(x-pi/3)= 1/2 ?

The general solution for sin(x-pi/3)= 1/2 is x=2pin+pi/2 ,x=2pin+(7pi)/6

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

Solution

sin (x - \frac{π}{3}) = \frac{1}{2}

Solution

x = 2 πn + \frac{π}{2}, x = 2 πn + \frac{7 π}{6}

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

Solution steps

sin (x - \frac{π}{3}) = \frac{1}{2}

General solutions for

sin (x - \frac{π}{3}) = \frac{1}{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}

x - \frac{π}{3} = \frac{π}{6} + 2 πn, x - \frac{π}{3} = \frac{5 π}{6} + 2 πn

x - \frac{π}{3} = \frac{π}{6} + 2 πn, x - \frac{π}{3} = \frac{5 π}{6} + 2 πn

Solve

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

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}

Solve

x - \frac{π}{3} = \frac{5 π}{6} + 2 πn : x = 2 πn + \frac{7 π}{6}

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

Move

\frac{π}{3}

to the right side

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

Add

\frac{π}{3}

to both sides

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

Simplify

x - \frac{π}{3} + \frac{π}{3} = \frac{5 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{5 π}{6} + 2 πn + \frac{π}{3} : 2 πn + \frac{7 π}{6}

\frac{5 π}{6} + 2 πn + \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{5 π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 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 2}{6} + \frac{5 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + 5 π}{6}

Add similar elements:

2 π + 5 π = 7 π

= 2 πn + \frac{7 π}{6}

x = 2 πn + \frac{7 π}{6}

x = 2 πn + \frac{7 π}{6}

x = 2 πn + \frac{7 π}{6}

x = 2 πn + \frac{π}{2}, x = 2 πn + \frac{7 π}{6}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x-pi/3)= 1/2 ?
The general solution for sin(x-pi/3)= 1/2 is x=2pin+pi/2 ,x=2pin+(7pi)/6