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

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

The general solution for sin((2θ)/3)= 1/2 ,0<= θ<= 2pi is θ= pi/4 ,θ=(5pi)/4

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sin((2θ)/3)= 1/2 ,0<= θ<= 2pi

Solution

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

Solution

θ = \frac{π}{4}, θ = \frac{5 π}{4}

Degrees

θ = 4 5^{\circ}, θ = 22 5^{\circ}

Solution steps

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

General solutions for

sin (\frac{2 θ}{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{2 θ}{3} = \frac{π}{6} + 2 πn, \frac{2 θ}{3} = \frac{5 π}{6} + 2 πn

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

Solve

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

\frac{2 θ}{3} = \frac{π}{6} + 2 πn

Multiply both sides by

3

\frac{2 θ}{3} = \frac{π}{6} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 \cdot 2 θ}{3} : 2 θ

\frac{3 \cdot 2 θ}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 θ}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 θ

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

2 θ = \frac{π}{2} + 6 πn

2 θ = \frac{π}{2} + 6 πn

2 θ = \frac{π}{2} + 6 πn

Divide both sides by

2

2 θ = \frac{π}{2} + 6 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{\frac{π}{2}}{2} + \frac{6 πn}{2}

Simplify

\frac{2 θ}{2} = \frac{\frac{π}{2}}{2} + \frac{6 πn}{2}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

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

\frac{\frac{π}{2}}{2} + \frac{6 πn}{2}

\frac{\frac{π}{2}}{2} = \frac{π}{4}

\frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

\frac{6 πn}{2} = 3 πn

\frac{6 πn}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 πn

= \frac{π}{4} + 3 πn

θ = \frac{π}{4} + 3 πn

θ = \frac{π}{4} + 3 πn

θ = \frac{π}{4} + 3 πn

Solve

\frac{2 θ}{3} = \frac{5 π}{6} + 2 πn : θ = \frac{5 π}{4} + 3 πn

\frac{2 θ}{3} = \frac{5 π}{6} + 2 πn

Multiply both sides by

3

\frac{2 θ}{3} = \frac{5 π}{6} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 \cdot 2 θ}{3} : 2 θ

\frac{3 \cdot 2 θ}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 θ}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 θ

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

2 θ = \frac{5 π}{2} + 6 πn

2 θ = \frac{5 π}{2} + 6 πn

2 θ = \frac{5 π}{2} + 6 πn

Divide both sides by

2

2 θ = \frac{5 π}{2} + 6 πn

Divide both sides by

2

\frac{2 θ}{2} = \frac{\frac{5 π}{2}}{2} + \frac{6 πn}{2}

Simplify

\frac{2 θ}{2} = \frac{\frac{5 π}{2}}{2} + \frac{6 πn}{2}

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

Simplify

\frac{\frac{5 π}{2}}{2} + \frac{6 πn}{2} : \frac{5 π}{4} + 3 πn

\frac{\frac{5 π}{2}}{2} + \frac{6 πn}{2}

\frac{\frac{5 π}{2}}{2} = \frac{5 π}{4}

\frac{\frac{5 π}{2}}{2}

Apply the fraction rule:

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

= \frac{5 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{5 π}{4}

\frac{6 πn}{2} = 3 πn

\frac{6 πn}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 πn

= \frac{5 π}{4} + 3 πn

θ = \frac{5 π}{4} + 3 πn

θ = \frac{5 π}{4} + 3 πn

θ = \frac{5 π}{4} + 3 πn

θ = \frac{π}{4} + 3 πn, θ = \frac{5 π}{4} + 3 πn

Solutions for the range

0 \leq θ \leq 2 π

θ = \frac{π}{4}, θ = \frac{5 π}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for sin((2θ)/3)= 1/2 ,0<= θ<= 2pi ?
The general solution for sin((2θ)/3)= 1/2 ,0<= θ<= 2pi is θ= pi/4 ,θ=(5pi)/4