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

Solution

cos (2 x + \frac{π}{3}) = \frac{3}{2}, 0 \leq x \leq 2 π

Solution

x = \frac{3 π}{4}, x = \frac{11 π}{12}, x = \frac{7 π}{4}, x = \frac{23 π}{12}

Degrees

x = 13 5^{\circ}, x = 16 5^{\circ}, x = 31 5^{\circ}, x = 34 5^{\circ}

Solution steps

cos (2 x + \frac{π}{3}) = \frac{3}{2}, 0 \leq x \leq 2 π

General solutions for

cos (2 x + \frac{π}{3}) = \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}

2 x + \frac{π}{3} = \frac{π}{6} + 2 πn, 2 x + \frac{π}{3} = \frac{11 π}{6} + 2 πn

2 x + \frac{π}{3} = \frac{π}{6} + 2 πn, 2 x + \frac{π}{3} = \frac{11 π}{6} + 2 πn

Solve

2 x + \frac{π}{3} = \frac{π}{6} + 2 πn : x = πn - \frac{π}{12}

2 x + \frac{π}{3} = \frac{π}{6} + 2 πn

Move

\frac{π}{3}

to the right side

2 x + \frac{π}{3} = \frac{π}{6} + 2 πn

Subtract

\frac{π}{3}

from both sides

2 x + \frac{π}{3} - \frac{π}{3} = \frac{π}{6} + 2 πn - \frac{π}{3}

Simplify

2 x + \frac{π}{3} - \frac{π}{3} = \frac{π}{6} + 2 πn - \frac{π}{3}

Simplify

2 x + \frac{π}{3} - \frac{π}{3} : 2 x

2 x + \frac{π}{3} - \frac{π}{3}

Add similar elements:

\frac{π}{3} - \frac{π}{3} = 0

= 2 x

Simplify

\frac{π}{6} + 2 πn - \frac{π}{3} : 2 πn - \frac{π}{6}

\frac{π}{6} + 2 πn - \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{6} - \frac{π}{3}

Least Common Multiplier of

6, 3 : 6

6, 3

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

6

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

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

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= \frac{π}{6} - \frac{π 2}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π - π 2}{6}

Add similar elements:

π - 2 π = - π

= \frac{- π}{6}

Apply the fraction rule:

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

= 2 πn - \frac{π}{6}

2 x = 2 πn - \frac{π}{6}

2 x = 2 πn - \frac{π}{6}

2 x = 2 πn - \frac{π}{6}

Divide both sides by

2

2 x = 2 πn - \frac{π}{6}

Divide both sides by

2

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

Simplify

\frac{2 x}{2} = \frac{2 πn}{2} - \frac{\frac{π}{6}}{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{π}{6}}{2} : πn - \frac{π}{12}

\frac{2 πn}{2} - \frac{\frac{π}{6}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

Apply the fraction rule:

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

= \frac{π}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{π}{12}

= πn - \frac{π}{12}

x = πn - \frac{π}{12}

x = πn - \frac{π}{12}

x = πn - \frac{π}{12}

Solve

2 x + \frac{π}{3} = \frac{11 π}{6} + 2 πn : x = πn + \frac{3 π}{4}

2 x + \frac{π}{3} = \frac{11 π}{6} + 2 πn

Move

\frac{π}{3}

to the right side

2 x + \frac{π}{3} = \frac{11 π}{6} + 2 πn

Subtract

\frac{π}{3}

from both sides

2 x + \frac{π}{3} - \frac{π}{3} = \frac{11 π}{6} + 2 πn - \frac{π}{3}

Simplify

2 x + \frac{π}{3} - \frac{π}{3} = \frac{11 π}{6} + 2 πn - \frac{π}{3}

Simplify

2 x + \frac{π}{3} - \frac{π}{3} : 2 x

2 x + \frac{π}{3} - \frac{π}{3}

Add similar elements:

\frac{π}{3} - \frac{π}{3} = 0

= 2 x

Simplify

\frac{11 π}{6} + 2 πn - \frac{π}{3} : 2 πn + \frac{3 π}{2}

\frac{11 π}{6} + 2 πn - \frac{π}{3}

Group like terms

= 2 πn - \frac{π}{3} + \frac{11 π}{6}

Least Common Multiplier of

3, 6 : 6

3, 6

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

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

3

6

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 6

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

6

For

\frac{π}{3} :

multiply the denominator and numerator by

2

\frac{π}{3} = \frac{π 2}{3 \cdot 2} = \frac{π 2}{6}

= - \frac{π 2}{6} + \frac{11 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 2 + 11 π}{6}

Add similar elements:

- 2 π + 11 π = 9 π

= \frac{9 π}{6}

Cancel the common factor:

3

= 2 πn + \frac{3 π}{2}

2 x = 2 πn + \frac{3 π}{2}

2 x = 2 πn + \frac{3 π}{2}

2 x = 2 πn + \frac{3 π}{2}

Divide both sides by

2

2 x = 2 πn + \frac{3 π}{2}

Divide both sides by

2

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

Simplify

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

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

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{3 π}{2}}{2} = \frac{3 π}{4}

\frac{\frac{3 π}{2}}{2}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4}

= πn + \frac{3 π}{4}

x = πn + \frac{3 π}{4}

x = πn + \frac{3 π}{4}

x = πn + \frac{3 π}{4}

x = πn - \frac{π}{12}, x = πn + \frac{3 π}{4}

Solutions for the range

0 \leq x \leq 2 π

x = \frac{3 π}{4}, x = \frac{11 π}{12}, x = \frac{7 π}{4}, x = \frac{23 π}{12}

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