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

Q: What is the general solution for sin(x/3)= 1/2 ,0<= x<= 2pi ?

The general solution for sin(x/3)= 1/2 ,0<= x<= 2pi is x= pi/2

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sin(x/3)= 1/2 ,0<= x<= 2pi

Solution

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

Solution

x = \frac{π}{2}

Degrees

x = 9 0^{\circ}

Solution steps

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

General solutions for

sin (\frac{x}{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}

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

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

Solve

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

\frac{x}{3} = \frac{π}{6} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{π}{6} + 2 πn

Multiply both sides by

3

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

Simplify

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

3 \cdot \frac{π}{6} + 3 \cdot 2 πn

3 \cdot \frac{π}{6} = \frac{π}{2}

3 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 3}{6}

Cancel the common factor:

3

= \frac{π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

x = \frac{π}{2} + 6 πn

Solve

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

\frac{x}{3} = \frac{5 π}{6} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{5 π}{6} + 2 πn

Multiply both sides by

3

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

Simplify

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

3 \cdot \frac{5 π}{6} + 3 \cdot 2 πn

3 \cdot \frac{5 π}{6} = \frac{5 π}{2}

3 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 3}{6}

Multiply the numbers:

5 \cdot 3 = 15

= \frac{15 π}{6}

Cancel the common factor:

3

= \frac{5 π}{2}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

x = \frac{5 π}{2} + 6 πn

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

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{2}

Popular Examples

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sin (θ) = - \frac{8}{17}

sin(θ/2)-cos(θ)=0

sin (\frac{θ}{2}) - cos (θ) = 0

12sec^2(x)-16=0

12 sec^{2} (x) - 16 = 0

cos(3x-pi/8)=(sqrt(2))/2

cos (3 x - \frac{π}{8}) = \frac{2}{2}

5(1-cos(θ))=sin^2(θ)

5 (1 - cos (θ)) = sin^{2} (θ)

Frequently Asked Questions (FAQ)

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