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

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

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cos(3θ)=-1/2

Solution

cos (3 θ) = - \frac{1}{2}

Solution

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

Degrees

θ = 4 0^{\circ} + 12 0^{\circ} n, θ = 8 0^{\circ} + 12 0^{\circ} n

Solution steps

cos (3 θ) = - \frac{1}{2}

General solutions for

cos (3 θ) = - \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}

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

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

Solve

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

3 θ = \frac{2 π}{3} + 2 πn

Divide both sides by

3

3 θ = \frac{2 π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

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

θ = \frac{2 π}{9} + \frac{2 πn}{3}

θ = \frac{2 π}{9} + \frac{2 πn}{3}

θ = \frac{2 π}{9} + \frac{2 πn}{3}

Solve

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

3 θ = \frac{4 π}{3} + 2 πn

Divide both sides by

3

3 θ = \frac{4 π}{3} + 2 πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

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

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

\frac{\frac{4 π}{3}}{3} = \frac{4 π}{9}

\frac{\frac{4 π}{3}}{3}

Apply the fraction rule:

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

= \frac{4 π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{4 π}{9}

= \frac{4 π}{9} + \frac{2 πn}{3}

θ = \frac{4 π}{9} + \frac{2 πn}{3}

θ = \frac{4 π}{9} + \frac{2 πn}{3}

θ = \frac{4 π}{9} + \frac{2 πn}{3}

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

Popular Examples

cos(x)= 5/6

cos (x) = \frac{5}{6}

1-tan^2(x)=0

1 - tan^{2} (x) = 0

cos(x)cos(3x)-sin(x)sin(3x)=0

cos (x) cos (3 x) - sin (x) sin (3 x) = 0

(cot(θ)-1)(sin(θ)-1)=0

(cot (θ) - 1) (sin (θ) - 1) = 0

cos(2x)=2sin^2(x)

cos (2 x) = 2 sin^{2} (x)

Frequently Asked Questions (FAQ)

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