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

The general solution for cos(x-(2pi)/3)= 1/2 is x=2pin+pi,x=2pin+(7pi)/3

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cos(x-(2pi)/3)= 1/2

Solution

cos (x - \frac{2 π}{3}) = \frac{1}{2}

Solution

x = 2 πn + π, x = 2 πn + \frac{7 π}{3}

Degrees

x = 18 0^{\circ} + 36 0^{\circ} n, x = 42 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (x - \frac{2 π}{3}) = \frac{1}{2}

General solutions for

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

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

Solve

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

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

Move

\frac{2 π}{3}

to the right side

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

Add

\frac{2 π}{3}

to both sides

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

Simplify

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

Simplify

x - \frac{2 π}{3} + \frac{2 π}{3} : x

x - \frac{2 π}{3} + \frac{2 π}{3}

Add similar elements:

- \frac{2 π}{3} + \frac{2 π}{3} = 0

= x

Simplify

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

\frac{π}{3} + 2 πn + \frac{2 π}{3}

Group like terms

= 2 πn + \frac{π}{3} + \frac{2 π}{3}

Combine the fractions

\frac{π}{3} + \frac{2 π}{3} : π

Apply rule

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

= \frac{π + 2 π}{3}

Add similar elements:

π + 2 π = 3 π

= \frac{3 π}{3}

Divide the numbers:

\frac{3}{3} = 1

= π

= 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn + π

Solve

x - \frac{2 π}{3} = \frac{5 π}{3} + 2 πn : x = 2 πn + \frac{7 π}{3}

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

Move

\frac{2 π}{3}

to the right side

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

Add

\frac{2 π}{3}

to both sides

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

Simplify

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

Simplify

x - \frac{2 π}{3} + \frac{2 π}{3} : x

x - \frac{2 π}{3} + \frac{2 π}{3}

Add similar elements:

- \frac{2 π}{3} + \frac{2 π}{3} = 0

= x

Simplify

\frac{5 π}{3} + 2 πn + \frac{2 π}{3} : 2 πn + \frac{7 π}{3}

\frac{5 π}{3} + 2 πn + \frac{2 π}{3}

Group like terms

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

Combine the fractions

\frac{5 π}{3} + \frac{2 π}{3} : \frac{7 π}{3}

Apply rule

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

= \frac{5 π + 2 π}{3}

Add similar elements:

5 π + 2 π = 7 π

= \frac{7 π}{3}

= 2 πn + \frac{7 π}{3}

x = 2 πn + \frac{7 π}{3}

x = 2 πn + \frac{7 π}{3}

x = 2 πn + \frac{7 π}{3}

x = 2 πn + π, x = 2 πn + \frac{7 π}{3}

Popular Examples

3sin(θ)=3

3 sin (θ) = 3

cos(2θ)+sin(θ)=0,0<= θ<2pi

cos (2 θ) + sin (θ) = 0, 0 \leq θ < 2 π

cos(pi/3 x-pi/4)=(sqrt(2))/2

cos (\frac{π}{3} x - \frac{π}{4}) = \frac{2}{2}

(sin(x))/1 = 3/5

\frac{sin ( x )}{1} = \frac{3}{5}

solvefor x,e^y=tan(x+pi/3)solve for

x, e^{y} = tan (x + \frac{π}{3})

Frequently Asked Questions (FAQ)

What is the general solution for cos(x-(2pi)/3)= 1/2 ?
The general solution for cos(x-(2pi)/3)= 1/2 is x=2pin+pi,x=2pin+(7pi)/3