What is the general solution for cos((2x-pi)/6)=((sqrt(3)))/2 ,0<x<2pi ?

Q: What is the general solution for cos((2x-pi)/6)=((sqrt(3)))/2 ,0<x<2pi ?

The general solution for cos((2x-pi)/6)=((sqrt(3)))/2 ,0<x<2pi is x=pi

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cos((2x-pi)/6)=((sqrt(3)))/2 ,0<x<2pi

Solution

cos (\frac{2 x - π}{6}) = \frac{( 3 )}{2}, 0 < x < 2 π

Solution

x = π

Degrees

x = 18 0^{\circ}

Solution steps

cos (\frac{2 x - π}{6}) = \frac{( 3 )}{2}, 0 < x < 2 π

General solutions for

cos (\frac{2 x - π}{6}) = \frac{3}{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{2 x - π}{6} = \frac{π}{6} + 2 πn, \frac{2 x - π}{6} = \frac{11 π}{6} + 2 πn

\frac{2 x - π}{6} = \frac{π}{6} + 2 πn, \frac{2 x - π}{6} = \frac{11 π}{6} + 2 πn

Solve

\frac{2 x - π}{6} = \frac{π}{6} + 2 πn : x = π + 6 πn

\frac{2 x - π}{6} = \frac{π}{6} + 2 πn

Multiply both sides by

6

\frac{2 x - π}{6} = \frac{π}{6} + 2 πn

Multiply both sides by

6

\frac{6 ( 2 x - π )}{6} = 6 \cdot \frac{π}{6} + 6 \cdot 2 πn

Simplify

\frac{6 ( 2 x - π )}{6} = 6 \cdot \frac{π}{6} + 6 \cdot 2 πn

Simplify

\frac{6 ( 2 x - π )}{6} : 2 x - π

\frac{6 ( 2 x - π )}{6}

Divide the numbers:

\frac{6}{6} = 1

= 2 x - π

Simplify

6 \cdot \frac{π}{6} + 6 \cdot 2 πn : π + 12 πn

6 \cdot \frac{π}{6} + 6 \cdot 2 πn

6 \cdot \frac{π}{6} = π

6 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 6}{6}

Cancel the common factor:

6

= π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= π + 12 πn

2 x - π = π + 12 πn

2 x - π = π + 12 πn

2 x - π = π + 12 πn

Move

π

to the right side

2 x - π = π + 12 πn

Add

π

to both sides

2 x - π + π = π + 12 πn + π

Simplify

2 x = 2 π + 12 πn

2 x = 2 π + 12 πn

Divide both sides by

2

2 x = 2 π + 12 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 π}{2} + \frac{12 πn}{2}

Simplify

\frac{2 x}{2} = \frac{2 π}{2} + \frac{12 πn}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 π}{2} + \frac{12 πn}{2} : π + 6 πn

\frac{2 π}{2} + \frac{12 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= π + \frac{12 πn}{2}

Divide the numbers:

\frac{12}{2} = 6

= π + 6 πn

x = π + 6 πn

x = π + 6 πn

x = π + 6 πn

Solve

\frac{2 x - π}{6} = \frac{11 π}{6} + 2 πn : x = 6 π + 6 πn

\frac{2 x - π}{6} = \frac{11 π}{6} + 2 πn

Multiply both sides by

6

\frac{2 x - π}{6} = \frac{11 π}{6} + 2 πn

Multiply both sides by

6

\frac{6 ( 2 x - π )}{6} = 6 \cdot \frac{11 π}{6} + 6 \cdot 2 πn

Simplify

\frac{6 ( 2 x - π )}{6} = 6 \cdot \frac{11 π}{6} + 6 \cdot 2 πn

Simplify

\frac{6 ( 2 x - π )}{6} : 2 x - π

\frac{6 ( 2 x - π )}{6}

Divide the numbers:

\frac{6}{6} = 1

= 2 x - π

Simplify

6 \cdot \frac{11 π}{6} + 6 \cdot 2 πn : 11 π + 12 πn

6 \cdot \frac{11 π}{6} + 6 \cdot 2 πn

6 \cdot \frac{11 π}{6} = 11 π

6 \cdot \frac{11 π}{6}

Multiply fractions:

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

= \frac{11 π 6}{6}

Cancel the common factor:

6

= 11 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 11 π + 12 πn

2 x - π = 11 π + 12 πn

2 x - π = 11 π + 12 πn

2 x - π = 11 π + 12 πn

Move

π

to the right side

2 x - π = 11 π + 12 πn

Add

π

to both sides

2 x - π + π = 11 π + 12 πn + π

Simplify

2 x = 12 π + 12 πn

2 x = 12 π + 12 πn

Divide both sides by

2

2 x = 12 π + 12 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{12 π}{2} + \frac{12 πn}{2}

Simplify

x = 6 π + 6 πn

x = 6 π + 6 πn

x = π + 6 πn, x = 6 π + 6 πn

Solutions for the range

0 < x < 2 π

x = π

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Frequently Asked Questions (FAQ)

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