What is the general solution for cos(2x-pi/(14))=0 ?

The general solution for cos(2x-pi/(14))=0 is x=pin+(2pi)/7 ,x=pin+(11pi)/(14)

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cos(2x-pi/(14))=0

Solution

cos (2 x - \frac{π}{14}) = 0

Solution

x = πn + \frac{2 π}{7}, x = πn + \frac{11 π}{14}

Degrees

x = 51.42857 \dots^{\circ} + 18 0^{\circ} n, x = 141.42857 \dots^{\circ} + 18 0^{\circ} n

Solution steps

cos (2 x - \frac{π}{14}) = 0

General solutions for

cos (2 x - \frac{π}{14}) = 0

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}

2 x - \frac{π}{14} = \frac{π}{2} + 2 πn, 2 x - \frac{π}{14} = \frac{3 π}{2} + 2 πn

2 x - \frac{π}{14} = \frac{π}{2} + 2 πn, 2 x - \frac{π}{14} = \frac{3 π}{2} + 2 πn

Solve

2 x - \frac{π}{14} = \frac{π}{2} + 2 πn : x = πn + \frac{2 π}{7}

2 x - \frac{π}{14} = \frac{π}{2} + 2 πn

Move

\frac{π}{14}

to the right side

2 x - \frac{π}{14} = \frac{π}{2} + 2 πn

Add

\frac{π}{14}

to both sides

2 x - \frac{π}{14} + \frac{π}{14} = \frac{π}{2} + 2 πn + \frac{π}{14}

Simplify

2 x - \frac{π}{14} + \frac{π}{14} = \frac{π}{2} + 2 πn + \frac{π}{14}

Simplify

2 x - \frac{π}{14} + \frac{π}{14} : 2 x

2 x - \frac{π}{14} + \frac{π}{14}

Add similar elements:

- \frac{π}{14} + \frac{π}{14} = 0

= 2 x

Simplify

\frac{π}{2} + 2 πn + \frac{π}{14} : 2 πn + \frac{4 π}{7}

\frac{π}{2} + 2 πn + \frac{π}{14}

Group like terms

= 2 πn + \frac{π}{2} + \frac{π}{14}

Least Common Multiplier of

2, 14 : 14

2, 14

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

14 : 2 \cdot 7

14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 7

Multiply each factor the greatest number of times it occurs in either

2

14

= 2 \cdot 7

Multiply the numbers:

2 \cdot 7 = 14

= 14

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

14

For

\frac{π}{2} :

multiply the denominator and numerator by

7

\frac{π}{2} = \frac{π 7}{2 \cdot 7} = \frac{π 7}{14}

= \frac{π 7}{14} + \frac{π}{14}

Since the denominators are equal, combine the fractions:

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

= \frac{π 7 + π}{14}

Add similar elements:

7 π + π = 8 π

= \frac{8 π}{14}

Cancel the common factor:

2

= 2 πn + \frac{4 π}{7}

2 x = 2 πn + \frac{4 π}{7}

2 x = 2 πn + \frac{4 π}{7}

2 x = 2 πn + \frac{4 π}{7}

Divide both sides by

2

2 x = 2 πn + \frac{4 π}{7}

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{\frac{4 π}{7}}{2}

Simplify

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{\frac{4 π}{7}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} + \frac{\frac{4 π}{7}}{2} : πn + \frac{2 π}{7}

\frac{2 πn}{2} + \frac{\frac{4 π}{7}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{4 π}{7}}{2} = \frac{2 π}{7}

\frac{\frac{4 π}{7}}{2}

Apply the fraction rule:

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

= \frac{4 π}{7 \cdot 2}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{4 π}{14}

Cancel the common factor:

2

= \frac{2 π}{7}

= πn + \frac{2 π}{7}

x = πn + \frac{2 π}{7}

x = πn + \frac{2 π}{7}

x = πn + \frac{2 π}{7}

Solve

2 x - \frac{π}{14} = \frac{3 π}{2} + 2 πn : x = πn + \frac{11 π}{14}

2 x - \frac{π}{14} = \frac{3 π}{2} + 2 πn

Move

\frac{π}{14}

to the right side

2 x - \frac{π}{14} = \frac{3 π}{2} + 2 πn

Add

\frac{π}{14}

to both sides

2 x - \frac{π}{14} + \frac{π}{14} = \frac{3 π}{2} + 2 πn + \frac{π}{14}

Simplify

2 x - \frac{π}{14} + \frac{π}{14} = \frac{3 π}{2} + 2 πn + \frac{π}{14}

Simplify

2 x - \frac{π}{14} + \frac{π}{14} : 2 x

2 x - \frac{π}{14} + \frac{π}{14}

Add similar elements:

- \frac{π}{14} + \frac{π}{14} = 0

= 2 x

Simplify

\frac{3 π}{2} + 2 πn + \frac{π}{14} : 2 πn + \frac{11 π}{7}

\frac{3 π}{2} + 2 πn + \frac{π}{14}

Group like terms

= 2 πn + \frac{3 π}{2} + \frac{π}{14}

Least Common Multiplier of

2, 14 : 14

2, 14

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

14 : 2 \cdot 7

14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 7

Multiply each factor the greatest number of times it occurs in either

2

14

= 2 \cdot 7

Multiply the numbers:

2 \cdot 7 = 14

= 14

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

14

For

\frac{3 π}{2} :

multiply the denominator and numerator by

7

\frac{3 π}{2} = \frac{3 π 7}{2 \cdot 7} = \frac{21 π}{14}

= \frac{21 π}{14} + \frac{π}{14}

Since the denominators are equal, combine the fractions:

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

= \frac{21 π + π}{14}

Add similar elements:

21 π + π = 22 π

= \frac{22 π}{14}

Cancel the common factor:

2

= 2 πn + \frac{11 π}{7}

2 x = 2 πn + \frac{11 π}{7}

2 x = 2 πn + \frac{11 π}{7}

2 x = 2 πn + \frac{11 π}{7}

Divide both sides by

2

2 x = 2 πn + \frac{11 π}{7}

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{\frac{11 π}{7}}{2}

Simplify

\frac{2 x}{2} = \frac{2 πn}{2} + \frac{\frac{11 π}{7}}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} + \frac{\frac{11 π}{7}}{2} : πn + \frac{11 π}{14}

\frac{2 πn}{2} + \frac{\frac{11 π}{7}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{11 π}{7}}{2} = \frac{11 π}{14}

\frac{\frac{11 π}{7}}{2}

Apply the fraction rule:

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

= \frac{11 π}{7 \cdot 2}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{11 π}{14}

= πn + \frac{11 π}{14}

x = πn + \frac{11 π}{14}

x = πn + \frac{11 π}{14}

x = πn + \frac{11 π}{14}

x = πn + \frac{2 π}{7}, x = πn + \frac{11 π}{14}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2x-pi/(14))=0 ?
The general solution for cos(2x-pi/(14))=0 is x=pin+(2pi)/7 ,x=pin+(11pi)/(14)