What is the general solution for 3-6cos(x+pi/3)=0 ?

The general solution for 3-6cos(x+pi/3)=0 is x=2pin,x=2pin+(4pi)/3

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3-6cos(x+pi/3)=0

Solution

3 - 6 cos (x + \frac{π}{3}) = 0

Solution

x = 2 πn, x = 2 πn + \frac{4 π}{3}

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

3 - 6 cos (x + \frac{π}{3}) = 0

Move

3

to the right side

3 - 6 cos (x + \frac{π}{3}) = 0

Subtract

3

from both sides

3 - 6 cos (x + \frac{π}{3}) - 3 = 0 - 3

Simplify

- 6 cos (x + \frac{π}{3}) = - 3

- 6 cos (x + \frac{π}{3}) = - 3

Divide both sides by

- 6

- 6 cos (x + \frac{π}{3}) = - 3

Divide both sides by

- 6

\frac{- 6 cos ( x + \frac{π}{3} )}{- 6} = \frac{- 3}{- 6}

Simplify

\frac{- 6 cos ( x + \frac{π}{3} )}{- 6} = \frac{- 3}{- 6}

Simplify

\frac{- 6 cos ( x + \frac{π}{3} )}{- 6} : cos (x + \frac{π}{3})

\frac{- 6 cos ( x + \frac{π}{3} )}{- 6}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{6 cos ( x + \frac{π}{3} )}{6}

Divide the numbers:

\frac{6}{6} = 1

= cos (x + \frac{π}{3})

Simplify

\frac{- 3}{- 6} : \frac{1}{2}

\frac{- 3}{- 6}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{3}{6}

Cancel the common factor:

3

= \frac{1}{2}

cos (x + \frac{π}{3}) = \frac{1}{2}

cos (x + \frac{π}{3}) = \frac{1}{2}

cos (x + \frac{π}{3}) = \frac{1}{2}

General solutions for

cos (x + \frac{π}{3}) = \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}

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

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

Solve

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

x + \frac{π}{3} = \frac{π}{3} + 2 πn

Subtract

\frac{π}{3}

from both sides

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

Simplify

x = 2 πn

Solve

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

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

Move

\frac{π}{3}

to the right side

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

Subtract

\frac{π}{3}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{3} - \frac{π}{3} : x

x + \frac{π}{3} - \frac{π}{3}

Add similar elements:

\frac{π}{3} - \frac{π}{3} = 0

= x

Simplify

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

\frac{5 π}{3} + 2 πn - \frac{π}{3}

Group like terms

= 2 πn - \frac{π}{3} + \frac{5 π}{3}

Combine the fractions

- \frac{π}{3} + \frac{5 π}{3} : \frac{4 π}{3}

Apply rule

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

= \frac{- π + 5 π}{3}

Add similar elements:

- π + 5 π = 4 π

= \frac{4 π}{3}

= 2 πn + \frac{4 π}{3}

x = 2 πn + \frac{4 π}{3}

x = 2 πn + \frac{4 π}{3}

x = 2 πn + \frac{4 π}{3}

x = 2 πn, x = 2 πn + \frac{4 π}{3}

Popular Examples

sin(7.07t)=0.5

sin (7.07 t) = 0.5

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

tan (x) = \frac{1}{3}, 0 \leq x < 2 π

-2sin(2x)=2pi

- 2 sin (2 x) = 2 π

arccos(θ)= 17/18

arccos (θ) = \frac{17}{18}

cos(θ)=-2/9

cos (θ) = - \frac{2}{9}

Frequently Asked Questions (FAQ)

What is the general solution for 3-6cos(x+pi/3)=0 ?
The general solution for 3-6cos(x+pi/3)=0 is x=2pin,x=2pin+(4pi)/3