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

The general solution for sin(pi/6 x)= 1/2 is x=1+12n,x=5+12n

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sin(pi/6 x)= 1/2

Solution

sin (\frac{π}{6} x) = \frac{1}{2}

Solution

x = 1 + 12 n, x = 5 + 12 n

Degrees

x = 57.29577 \dots^{\circ} + 687.54935 \dots^{\circ} n, x = 286.47889 \dots^{\circ} + 687.54935 \dots^{\circ} n

Solution steps

sin (\frac{π}{6} x) = \frac{1}{2}

General solutions for

sin (\frac{π}{6} x) = \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}

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

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

Solve

\frac{π}{6} x = \frac{π}{6} + 2 πn : x = 1 + 12 n

\frac{π}{6} x = \frac{π}{6} + 2 πn

Multiply both sides by

6

\frac{π}{6} x = \frac{π}{6} + 2 πn

Multiply both sides by

6

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

Simplify

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

Simplify

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiply fractions:

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

= \frac{6 π}{6} x

Cancel the common factor:

6

= x π

Simplify

6 \cdot \frac{π}{6} + 6 \cdot 2 πn : π + 12 πn

6 \cdot \frac{π}{6} + 6 \cdot 2 πn

6 \cdot \frac{π}{6} = π

6 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 6}{6}

Cancel the common factor:

6

= π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= π + 12 πn

π x = π + 12 πn

π x = π + 12 πn

π x = π + 12 πn

Divide both sides by

π

π x = π + 12 πn

Divide both sides by

π

\frac{π x}{π} = \frac{π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} = \frac{π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{π}{π} + \frac{12 πn}{π} : 1 + 12 n

\frac{π}{π} + \frac{12 πn}{π}

Apply rule

\frac{a}{a} = 1

\frac{π}{π} = 1

= 1 + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 1 + 12 n

x = 1 + 12 n

x = 1 + 12 n

x = 1 + 12 n

Solve

\frac{π}{6} x = \frac{5 π}{6} + 2 πn : x = 5 + 12 n

\frac{π}{6} x = \frac{5 π}{6} + 2 πn

Multiply both sides by

6

\frac{π}{6} x = \frac{5 π}{6} + 2 πn

Multiply both sides by

6

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

Simplify

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

Simplify

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiply fractions:

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

= \frac{6 π}{6} x

Cancel the common factor:

6

= x π

Simplify

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

6 \cdot \frac{5 π}{6} + 6 \cdot 2 πn

6 \cdot \frac{5 π}{6} = 5 π

6 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 6}{6}

Cancel the common factor:

6

= 5 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 5 π + 12 πn

π x = 5 π + 12 πn

π x = 5 π + 12 πn

π x = 5 π + 12 πn

Divide both sides by

π

π x = 5 π + 12 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{5 π}{π} + \frac{12 πn}{π} : 5 + 12 n

\frac{5 π}{π} + \frac{12 πn}{π}

Cancel

\frac{5 π}{π} : 5

\frac{5 π}{π}

Cancel the common factor:

π

= 5

= 5 + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 5 + 12 n

x = 5 + 12 n

x = 5 + 12 n

x = 5 + 12 n

x = 1 + 12 n, x = 5 + 12 n

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Frequently Asked Questions (FAQ)

What is the general solution for sin(pi/6 x)= 1/2 ?
The general solution for sin(pi/6 x)= 1/2 is x=1+12n,x=5+12n