What is the general solution for sin(1/2 x+pi/6)=(sqrt(3))/2 ?

The general solution for sin(1/2 x+pi/6)=(sqrt(3))/2 is x=4pin+pi/3 ,x=4pin+pi

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Popular Trigonometry >

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

Solution

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

Solution

x = 4 πn + \frac{π}{3}, x = 4 πn + π

Degrees

x = 6 0^{\circ} + 72 0^{\circ} n, x = 18 0^{\circ} + 72 0^{\circ} n

Solution steps

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

General solutions for

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

\frac{1}{2} x + \frac{π}{6} = \frac{π}{3} + 2 πn, \frac{1}{2} x + \frac{π}{6} = \frac{2 π}{3} + 2 πn

Solve

\frac{1}{2} x + \frac{π}{6} = \frac{π}{3} + 2 πn : x = 4 πn + \frac{π}{3}

\frac{1}{2} x + \frac{π}{6} = \frac{π}{3} + 2 πn

Move

\frac{π}{6}

to the right side

\frac{1}{2} x + \frac{π}{6} = \frac{π}{3} + 2 πn

Subtract

\frac{π}{6}

from both sides

\frac{1}{2} x + \frac{π}{6} - \frac{π}{6} = \frac{π}{3} + 2 πn - \frac{π}{6}

Simplify

\frac{1}{2} x + \frac{π}{6} - \frac{π}{6} = \frac{π}{3} + 2 πn - \frac{π}{6}

Simplify

\frac{1}{2} x + \frac{π}{6} - \frac{π}{6} : \frac{1}{2} x

\frac{1}{2} x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= \frac{1}{2} x

Simplify

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

\frac{π}{3} + 2 πn - \frac{π}{6}

Group like terms

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

Add similar elements:

2 π - π = π

= 2 πn + \frac{π}{6}

\frac{1}{2} x = 2 πn + \frac{π}{6}

\frac{1}{2} x = 2 πn + \frac{π}{6}

\frac{1}{2} x = 2 πn + \frac{π}{6}

Multiply both sides by

2

\frac{1}{2} x = 2 πn + \frac{π}{6}

Multiply both sides by

2

2 \cdot \frac{1}{2} x = 2 \cdot 2 πn + 2 \cdot \frac{π}{6}

Simplify

2 \cdot \frac{1}{2} x = 2 \cdot 2 πn + 2 \cdot \frac{π}{6}

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

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

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

2 \cdot 2 πn + 2 \cdot \frac{π}{6} : 4 πn + \frac{π}{3}

2 \cdot 2 πn + 2 \cdot \frac{π}{6}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

= 4 πn + \frac{π}{3}

x = 4 πn + \frac{π}{3}

x = 4 πn + \frac{π}{3}

x = 4 πn + \frac{π}{3}

Solve

\frac{1}{2} x + \frac{π}{6} = \frac{2 π}{3} + 2 πn : x = 4 πn + π

\frac{1}{2} x + \frac{π}{6} = \frac{2 π}{3} + 2 πn

Move

\frac{π}{6}

to the right side

\frac{1}{2} x + \frac{π}{6} = \frac{2 π}{3} + 2 πn

Subtract

\frac{π}{6}

from both sides

\frac{1}{2} x + \frac{π}{6} - \frac{π}{6} = \frac{2 π}{3} + 2 πn - \frac{π}{6}

Simplify

\frac{1}{2} x + \frac{π}{6} - \frac{π}{6} = \frac{2 π}{3} + 2 πn - \frac{π}{6}

Simplify

\frac{1}{2} x + \frac{π}{6} - \frac{π}{6} : \frac{1}{2} x

\frac{1}{2} x + \frac{π}{6} - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} = 0

= \frac{1}{2} x

Simplify

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

\frac{2 π}{3} + 2 πn - \frac{π}{6}

Group like terms

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

multiply the denominator and numerator by

2

\frac{2 π}{3} = \frac{2 π 2}{3 \cdot 2} = \frac{4 π}{6}

= - \frac{π}{6} + \frac{4 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π + 4 π}{6}

Add similar elements:

- π + 4 π = 3 π

= \frac{3 π}{6}

Cancel the common factor:

3

= 2 πn + \frac{π}{2}

\frac{1}{2} x = 2 πn + \frac{π}{2}

\frac{1}{2} x = 2 πn + \frac{π}{2}

\frac{1}{2} x = 2 πn + \frac{π}{2}

Multiply both sides by

2

\frac{1}{2} x = 2 πn + \frac{π}{2}

Multiply both sides by

2

2 \cdot \frac{1}{2} x = 2 \cdot 2 πn + 2 \cdot \frac{π}{2}

Simplify

2 \cdot \frac{1}{2} x = 2 \cdot 2 πn + 2 \cdot \frac{π}{2}

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

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

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

2 \cdot 2 πn + 2 \cdot \frac{π}{2} : 4 πn + π

2 \cdot 2 πn + 2 \cdot \frac{π}{2}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

= 4 πn + π

x = 4 πn + π

x = 4 πn + π

x = 4 πn + π

x = 4 πn + \frac{π}{3}, x = 4 πn + π

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(1/2 x+pi/6)=(sqrt(3))/2 ?
The general solution for sin(1/2 x+pi/6)=(sqrt(3))/2 is x=4pin+pi/3 ,x=4pin+pi