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

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

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

Solution

cos (x - \frac{π}{3}) = \frac{1}{2}

Solution

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

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} + 36 0^{\circ} n

Solution steps

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 = \frac{2 π}{3} + 2 πn

x - \frac{π}{3} = \frac{π}{3} + 2 πn

Move

\frac{π}{3}

to the right side

x - \frac{π}{3} = \frac{π}{3} + 2 πn

Add

\frac{π}{3}

to both sides

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

Simplify

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

\frac{π}{3} + 2 πn + \frac{π}{3}

Group like terms

= \frac{π}{3} + \frac{π}{3} + 2 πn

Combine the fractions

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

Apply rule

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

= \frac{π + π}{3}

Add similar elements:

π + π = 2 π

= \frac{2 π}{3}

= \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn

Solve

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

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

Add

\frac{π}{3}

to 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 + 2 π

\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} : 2 π

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 = 6 π

= \frac{6 π}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 π

= 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

x = 2 πn + 2 π

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

Popular Examples

tan(x)= 7/5

tan (x) = \frac{7}{5}

-2cos(2x)=0

- 2 cos (2 x) = 0

sin(7x)-1=0

sin (7 x) - 1 = 0

cos(x)(2sin(x)-1)=0

cos (x) (2 sin (x) - 1) = 0

tan(1/2 x)=3cos(1/2 x)

tan (\frac{1}{2} x) = 3 cos (\frac{1}{2} x)

Frequently Asked Questions (FAQ)

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