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

The general solution for cos(x/2+pi/3)= 1/(sqrt(2)) is x=4pin-pi/6 ,x=4pin+(17pi)/6

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cos(x/2+pi/3)= 1/(sqrt(2))

Solution

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

Solution

x = 4 πn - \frac{π}{6}, x = 4 πn + \frac{17 π}{6}

Radians

x = - \frac{π}{6} + 4 πn, x = \frac{17 π}{6} + 4 πn

Solution steps

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

Simplify

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

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

General solutions for

cos (\frac{x}{2} + \frac{π}{3}) = \frac{2}{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{x}{2} + \frac{π}{3} = \frac{π}{4} + 2 πn, \frac{x}{2} + \frac{π}{3} = \frac{7 π}{4} + 2 πn

\frac{x}{2} + \frac{π}{3} = \frac{π}{4} + 2 πn, \frac{x}{2} + \frac{π}{3} = \frac{7 π}{4} + 2 πn

Solve

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

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

Move

\frac{π}{3}

to the right side

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

Subtract

\frac{π}{3}

from both sides

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

Simplify

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

Simplify

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

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

Add similar elements:

\frac{π}{3} - \frac{π}{3} = 0

= \frac{x}{2}

Simplify

\frac{π}{4} + 2 πn - \frac{π}{3} : 2 πn - \frac{π}{12}

\frac{π}{4} + 2 πn - \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{4} - \frac{π}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

4

3

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{π}{3} :

multiply the denominator and numerator by

4

\frac{π}{3} = \frac{π 4}{3 \cdot 4} = \frac{π 4}{12}

= \frac{π 3}{12} - \frac{π 4}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π 4}{12}

Add similar elements:

3 π - 4 π = - π

= \frac{- π}{12}

Apply the fraction rule:

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

= 2 πn - \frac{π}{12}

\frac{x}{2} = 2 πn - \frac{π}{12}

\frac{x}{2} = 2 πn - \frac{π}{12}

\frac{x}{2} = 2 πn - \frac{π}{12}

Multiply both sides by

2

\frac{x}{2} = 2 πn - \frac{π}{12}

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 πn - 2 \cdot \frac{π}{12}

Simplify

\frac{2 x}{2} = 2 \cdot 2 πn - 2 \cdot \frac{π}{12}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot 2 πn - 2 \cdot \frac{π}{12} : 4 πn - \frac{π}{6}

2 \cdot 2 πn - 2 \cdot \frac{π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{π}{12} = \frac{π}{6}

2 \cdot \frac{π}{12}

Multiply fractions:

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

= \frac{π 2}{12}

Cancel the common factor:

2

= \frac{π}{6}

= 4 πn - \frac{π}{6}

x = 4 πn - \frac{π}{6}

x = 4 πn - \frac{π}{6}

x = 4 πn - \frac{π}{6}

Solve

\frac{x}{2} + \frac{π}{3} = \frac{7 π}{4} + 2 πn : x = 4 πn + \frac{17 π}{6}

\frac{x}{2} + \frac{π}{3} = \frac{7 π}{4} + 2 πn

Move

\frac{π}{3}

to the right side

\frac{x}{2} + \frac{π}{3} = \frac{7 π}{4} + 2 πn

Subtract

\frac{π}{3}

from both sides

\frac{x}{2} + \frac{π}{3} - \frac{π}{3} = \frac{7 π}{4} + 2 πn - \frac{π}{3}

Simplify

\frac{x}{2} + \frac{π}{3} - \frac{π}{3} = \frac{7 π}{4} + 2 πn - \frac{π}{3}

Simplify

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

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

Add similar elements:

\frac{π}{3} - \frac{π}{3} = 0

= \frac{x}{2}

Simplify

\frac{7 π}{4} + 2 πn - \frac{π}{3} : 2 πn + \frac{17 π}{12}

\frac{7 π}{4} + 2 πn - \frac{π}{3}

Group like terms

= 2 πn - \frac{π}{3} + \frac{7 π}{4}

Least Common Multiplier of

3, 4 : 12

3, 4

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

3

4

= 3 \cdot 2 \cdot 2

Multiply the numbers:

3 \cdot 2 \cdot 2 = 12

= 12

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

12

For

\frac{π}{3} :

multiply the denominator and numerator by

4

\frac{π}{3} = \frac{π 4}{3 \cdot 4} = \frac{π 4}{12}

For

\frac{7 π}{4} :

multiply the denominator and numerator by

3

\frac{7 π}{4} = \frac{7 π 3}{4 \cdot 3} = \frac{21 π}{12}

= - \frac{π 4}{12} + \frac{21 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 4 + 21 π}{12}

Add similar elements:

- 4 π + 21 π = 17 π

= 2 πn + \frac{17 π}{12}

\frac{x}{2} = 2 πn + \frac{17 π}{12}

\frac{x}{2} = 2 πn + \frac{17 π}{12}

\frac{x}{2} = 2 πn + \frac{17 π}{12}

Multiply both sides by

2

\frac{x}{2} = 2 πn + \frac{17 π}{12}

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 πn + 2 \cdot \frac{17 π}{12}

Simplify

\frac{2 x}{2} = 2 \cdot 2 πn + 2 \cdot \frac{17 π}{12}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot 2 πn + 2 \cdot \frac{17 π}{12} : 4 πn + \frac{17 π}{6}

2 \cdot 2 πn + 2 \cdot \frac{17 π}{12}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

2 \cdot \frac{17 π}{12} = \frac{17 π}{6}

2 \cdot \frac{17 π}{12}

Multiply fractions:

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

= \frac{17 π 2}{12}

Multiply the numbers:

17 \cdot 2 = 34

= \frac{34 π}{12}

Cancel the common factor:

2

= \frac{17 π}{6}

= 4 πn + \frac{17 π}{6}

x = 4 πn + \frac{17 π}{6}

x = 4 πn + \frac{17 π}{6}

x = 4 πn + \frac{17 π}{6}

x = 4 πn - \frac{π}{6}, x = 4 πn + \frac{17 π}{6}

Popular Examples

sinh(x)=(sqrt(2))/2

sinh (x) = \frac{2}{2}

sin^2(2x)=cos^2(x)

sin^{2} (2 x) = cos^{2} (x)

-2sin(2x)sin(x)=sin(2x)

- 2 sin (2 x) sin (x) = sin (2 x)

2sin(θ)-sqrt(2)=0,0<= θ<= 2pi

2 sin (θ) - 2 = 0, 0 \leq θ \leq 2 π

sin(θ)-0.2cos(θ)=(6.25)/(9.8)

sin (θ) - 0.2 cos (θ) = \frac{6.25}{9.8}

Frequently Asked Questions (FAQ)

What is the general solution for cos(x/2+pi/3)= 1/(sqrt(2)) ?
The general solution for cos(x/2+pi/3)= 1/(sqrt(2)) is x=4pin-pi/6 ,x=4pin+(17pi)/6