What is the general solution for cos((3θ)/2)= 1/2 ?

The general solution for cos((3θ)/2)= 1/2 is θ=(2pi)/9+(4pin)/3 ,θ=(10pi)/9+(4pin)/3

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cos((3θ)/2)= 1/2

Solution

cos (\frac{3 θ}{2}) = \frac{1}{2}

Solution

θ = \frac{2 π}{9} + \frac{4 πn}{3}, θ = \frac{10 π}{9} + \frac{4 πn}{3}

Degrees

θ = 4 0^{\circ} + 24 0^{\circ} n, θ = 20 0^{\circ} + 24 0^{\circ} n

Solution steps

cos (\frac{3 θ}{2}) = \frac{1}{2}

General solutions for

cos (\frac{3 θ}{2}) = \frac{1}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

Solve

\frac{3 θ}{2} = \frac{π}{3} + 2 πn : θ = \frac{2 π}{9} + \frac{4 πn}{3}

\frac{3 θ}{2} = \frac{π}{3} + 2 πn

Multiply both sides by

2

\frac{3 θ}{2} = \frac{π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 3 θ}{2} : 3 θ

\frac{2 \cdot 3 θ}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 θ}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 θ

Simplify

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

2 \cdot \frac{π}{3} + 2 \cdot 2 πn

Multiply

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 2}{3}

= \frac{2 π}{3} + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 π}{3} + 4 πn

3 θ = \frac{2 π}{3} + 4 πn

3 θ = \frac{2 π}{3} + 4 πn

3 θ = \frac{2 π}{3} + 4 πn

Divide both sides by

3

3 θ = \frac{2 π}{3} + 4 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

\frac{\frac{2 π}{3}}{3} + \frac{4 πn}{3} : \frac{2 π}{9} + \frac{4 πn}{3}

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

\frac{\frac{2 π}{3}}{3} = \frac{2 π}{9}

\frac{\frac{2 π}{3}}{3}

Apply the fraction rule:

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

= \frac{2 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{2 π}{9}

= \frac{2 π}{9} + \frac{4 πn}{3}

θ = \frac{2 π}{9} + \frac{4 πn}{3}

θ = \frac{2 π}{9} + \frac{4 πn}{3}

θ = \frac{2 π}{9} + \frac{4 πn}{3}

Solve

\frac{3 θ}{2} = \frac{5 π}{3} + 2 πn : θ = \frac{10 π}{9} + \frac{4 πn}{3}

\frac{3 θ}{2} = \frac{5 π}{3} + 2 πn

Multiply both sides by

2

\frac{3 θ}{2} = \frac{5 π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 3 θ}{2} : 3 θ

\frac{2 \cdot 3 θ}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 θ}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 θ

Simplify

2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn : \frac{10 π}{3} + 4 πn

2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{3} = \frac{10 π}{3}

2 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 2}{3}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{10 π}{3} + 4 πn

3 θ = \frac{10 π}{3} + 4 πn

3 θ = \frac{10 π}{3} + 4 πn

3 θ = \frac{10 π}{3} + 4 πn

Divide both sides by

3

3 θ = \frac{10 π}{3} + 4 πn

Divide both sides by

3

\frac{3 θ}{3} = \frac{\frac{10 π}{3}}{3} + \frac{4 πn}{3}

Simplify

\frac{3 θ}{3} = \frac{\frac{10 π}{3}}{3} + \frac{4 πn}{3}

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

\frac{\frac{10 π}{3}}{3} + \frac{4 πn}{3} : \frac{10 π}{9} + \frac{4 πn}{3}

\frac{\frac{10 π}{3}}{3} + \frac{4 πn}{3}

\frac{\frac{10 π}{3}}{3} = \frac{10 π}{9}

\frac{\frac{10 π}{3}}{3}

Apply the fraction rule:

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

= \frac{10 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{10 π}{9}

= \frac{10 π}{9} + \frac{4 πn}{3}

θ = \frac{10 π}{9} + \frac{4 πn}{3}

θ = \frac{10 π}{9} + \frac{4 πn}{3}

θ = \frac{10 π}{9} + \frac{4 πn}{3}

θ = \frac{2 π}{9} + \frac{4 πn}{3}, θ = \frac{10 π}{9} + \frac{4 πn}{3}

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Frequently Asked Questions (FAQ)

What is the general solution for cos((3θ)/2)= 1/2 ?
The general solution for cos((3θ)/2)= 1/2 is θ=(2pi)/9+(4pin)/3 ,θ=(10pi)/9+(4pin)/3