What is the general solution for sin(1/3 x+(3pi)/2)=0 ?

The general solution for sin(1/3 x+(3pi)/2)=0 is x=6pin-(9pi)/2 ,x=3pi+6pin-(9pi)/2

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sin(1/3 x+(3pi)/2)=0

Solution

sin (\frac{1}{3} x + \frac{3 π}{2}) = 0

Solution

x = 6 πn - \frac{9 π}{2}, x = 3 π + 6 πn - \frac{9 π}{2}

Degrees

x = - 81 0^{\circ} + 108 0^{\circ} n, x = - 27 0^{\circ} + 108 0^{\circ} n

Solution steps

sin (\frac{1}{3} x + \frac{3 π}{2}) = 0

General solutions for

sin (\frac{1}{3} x + \frac{3 π}{2}) = 0

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}{3} x + \frac{3 π}{2} = 0 + 2 πn, \frac{1}{3} x + \frac{3 π}{2} = π + 2 πn

\frac{1}{3} x + \frac{3 π}{2} = 0 + 2 πn, \frac{1}{3} x + \frac{3 π}{2} = π + 2 πn

Solve

\frac{1}{3} x + \frac{3 π}{2} = 0 + 2 πn : x = 6 πn - \frac{9 π}{2}

\frac{1}{3} x + \frac{3 π}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{1}{3} x + \frac{3 π}{2} = 2 πn

Move

\frac{3 π}{2}

to the right side

\frac{1}{3} x + \frac{3 π}{2} = 2 πn

Subtract

\frac{3 π}{2}

from both sides

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

Simplify

\frac{1}{3} x = 2 πn - \frac{3 π}{2}

\frac{1}{3} x = 2 πn - \frac{3 π}{2}

Multiply both sides by

3

\frac{1}{3} x = 2 πn - \frac{3 π}{2}

Multiply both sides by

3

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

Simplify

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

Simplify

3 \cdot \frac{1}{3} x : x

3 \cdot \frac{1}{3} x

Multiply fractions:

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

= \frac{1 \cdot 3}{3} x

Cancel the common factor:

3

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

3 \cdot 2 πn - 3 \cdot \frac{3 π}{2} : 6 πn - \frac{9 π}{2}

3 \cdot 2 πn - 3 \cdot \frac{3 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

3 \cdot \frac{3 π}{2} = \frac{9 π}{2}

3 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 3}{2}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{9 π}{2}

= 6 πn - \frac{9 π}{2}

x = 6 πn - \frac{9 π}{2}

x = 6 πn - \frac{9 π}{2}

x = 6 πn - \frac{9 π}{2}

Solve

\frac{1}{3} x + \frac{3 π}{2} = π + 2 πn : x = 3 π + 6 πn - \frac{9 π}{2}

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

Move

\frac{3 π}{2}

to the right side

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

Subtract

\frac{3 π}{2}

from both sides

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

Simplify

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

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

Multiply both sides by

3

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

Multiply both sides by

3

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

Simplify

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

Simplify

3 \cdot \frac{1}{3} x : x

3 \cdot \frac{1}{3} x

Multiply fractions:

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

= \frac{1 \cdot 3}{3} x

Cancel the common factor:

3

= x \cdot 1

Multiply:

x \cdot 1 = x

= x

Simplify

3 π + 3 \cdot 2 πn - 3 \cdot \frac{3 π}{2} : 3 π + 6 πn - \frac{9 π}{2}

3 π + 3 \cdot 2 πn - 3 \cdot \frac{3 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

3 \cdot \frac{3 π}{2} = \frac{9 π}{2}

3 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 3}{2}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{9 π}{2}

= 3 π + 6 πn - \frac{9 π}{2}

x = 3 π + 6 πn - \frac{9 π}{2}

x = 3 π + 6 πn - \frac{9 π}{2}

x = 3 π + 6 πn - \frac{9 π}{2}

x = 6 πn - \frac{9 π}{2}, x = 3 π + 6 πn - \frac{9 π}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(1/3 x+(3pi)/2)=0 ?
The general solution for sin(1/3 x+(3pi)/2)=0 is x=6pin-(9pi)/2 ,x=3pi+6pin-(9pi)/2