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

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

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sin(x/3)=(sqrt(3))/2

Solution

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

Solution

x = π + 6 πn, x = 2 π + 6 πn

Degrees

x = 18 0^{\circ} + 108 0^{\circ} n, x = 36 0^{\circ} + 108 0^{\circ} n

Solution steps

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

General solutions for

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

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

Solve

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

\frac{x}{3} = \frac{π}{3} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{π}{3} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

3 \cdot \frac{π}{3} + 3 \cdot 2 πn

3 \cdot \frac{π}{3} = π

3 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 3}{3}

Cancel the common factor:

3

= π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= π + 6 πn

x = π + 6 πn

x = π + 6 πn

x = π + 6 πn

Solve

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

\frac{x}{3} = \frac{2 π}{3} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{2 π}{3} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

3 \cdot \frac{2 π}{3} + 3 \cdot 2 πn

3 \cdot \frac{2 π}{3} = 2 π

3 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 3}{3}

Cancel the common factor:

3

= 2 π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= 2 π + 6 πn

x = 2 π + 6 πn

x = 2 π + 6 πn

x = 2 π + 6 πn

x = π + 6 πn, x = 2 π + 6 πn

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Frequently Asked Questions (FAQ)

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