What is the general solution for cos(θ/3-pi/4)= 1/2 ?

The general solution for cos(θ/3-pi/4)= 1/2 is θ=6pin+(7pi)/4 ,θ=6pin+(23pi)/4

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cos(θ/3-pi/4)= 1/2

Solution

cos (\frac{θ}{3} - \frac{π}{4}) = \frac{1}{2}

Solution

θ = 6 πn + \frac{7 π}{4}, θ = 6 πn + \frac{23 π}{4}

Degrees

θ = 31 5^{\circ} + 108 0^{\circ} n, θ = 103 5^{\circ} + 108 0^{\circ} n

Solution steps

cos (\frac{θ}{3} - \frac{π}{4}) = \frac{1}{2}

General solutions for

cos (\frac{θ}{3} - \frac{π}{4}) = \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}

\frac{θ}{3} - \frac{π}{4} = \frac{π}{3} + 2 πn, \frac{θ}{3} - \frac{π}{4} = \frac{5 π}{3} + 2 πn

\frac{θ}{3} - \frac{π}{4} = \frac{π}{3} + 2 πn, \frac{θ}{3} - \frac{π}{4} = \frac{5 π}{3} + 2 πn

Solve

\frac{θ}{3} - \frac{π}{4} = \frac{π}{3} + 2 πn : θ = 6 πn + \frac{7 π}{4}

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

Move

\frac{π}{4}

to the right side

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

Add

\frac{π}{4}

to both sides

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

Simplify

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

Simplify

\frac{θ}{3} - \frac{π}{4} + \frac{π}{4} : \frac{θ}{3}

\frac{θ}{3} - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= \frac{θ}{3}

Simplify

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

\frac{π}{3} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{3} + \frac{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{π}{4} :

multiply the denominator and numerator by

3

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

= \frac{π 4}{12} + \frac{π 3}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 4 + π 3}{12}

Add similar elements:

4 π + 3 π = 7 π

= 2 πn + \frac{7 π}{12}

\frac{θ}{3} = 2 πn + \frac{7 π}{12}

\frac{θ}{3} = 2 πn + \frac{7 π}{12}

\frac{θ}{3} = 2 πn + \frac{7 π}{12}

Multiply both sides by

3

\frac{θ}{3} = 2 πn + \frac{7 π}{12}

Multiply both sides by

3

\frac{3 θ}{3} = 3 \cdot 2 πn + 3 \cdot \frac{7 π}{12}

Simplify

\frac{3 θ}{3} = 3 \cdot 2 πn + 3 \cdot \frac{7 π}{12}

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

3 \cdot 2 πn + 3 \cdot \frac{7 π}{12} : 6 πn + \frac{7 π}{4}

3 \cdot 2 πn + 3 \cdot \frac{7 π}{12}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

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

3 \cdot \frac{7 π}{12}

Multiply fractions:

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

= \frac{7 π 3}{12}

Multiply the numbers:

7 \cdot 3 = 21

= \frac{21 π}{12}

Cancel the common factor:

3

= \frac{7 π}{4}

= 6 πn + \frac{7 π}{4}

θ = 6 πn + \frac{7 π}{4}

θ = 6 πn + \frac{7 π}{4}

θ = 6 πn + \frac{7 π}{4}

Solve

\frac{θ}{3} - \frac{π}{4} = \frac{5 π}{3} + 2 πn : θ = 6 πn + \frac{23 π}{4}

\frac{θ}{3} - \frac{π}{4} = \frac{5 π}{3} + 2 πn

Move

\frac{π}{4}

to the right side

\frac{θ}{3} - \frac{π}{4} = \frac{5 π}{3} + 2 πn

Add

\frac{π}{4}

to both sides

\frac{θ}{3} - \frac{π}{4} + \frac{π}{4} = \frac{5 π}{3} + 2 πn + \frac{π}{4}

Simplify

\frac{θ}{3} - \frac{π}{4} + \frac{π}{4} = \frac{5 π}{3} + 2 πn + \frac{π}{4}

Simplify

\frac{θ}{3} - \frac{π}{4} + \frac{π}{4} : \frac{θ}{3}

\frac{θ}{3} - \frac{π}{4} + \frac{π}{4}

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= \frac{θ}{3}

Simplify

\frac{5 π}{3} + 2 πn + \frac{π}{4} : 2 πn + \frac{23 π}{12}

\frac{5 π}{3} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{4} + \frac{5 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{5 π}{3} :

multiply the denominator and numerator by

4

\frac{5 π}{3} = \frac{5 π 4}{3 \cdot 4} = \frac{20 π}{12}

= \frac{π 3}{12} + \frac{20 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + 20 π}{12}

Add similar elements:

3 π + 20 π = 23 π

= 2 πn + \frac{23 π}{12}

\frac{θ}{3} = 2 πn + \frac{23 π}{12}

\frac{θ}{3} = 2 πn + \frac{23 π}{12}

\frac{θ}{3} = 2 πn + \frac{23 π}{12}

Multiply both sides by

3

\frac{θ}{3} = 2 πn + \frac{23 π}{12}

Multiply both sides by

3

\frac{3 θ}{3} = 3 \cdot 2 πn + 3 \cdot \frac{23 π}{12}

Simplify

\frac{3 θ}{3} = 3 \cdot 2 πn + 3 \cdot \frac{23 π}{12}

Simplify

\frac{3 θ}{3} : θ

\frac{3 θ}{3}

Divide the numbers:

\frac{3}{3} = 1

= θ

Simplify

3 \cdot 2 πn + 3 \cdot \frac{23 π}{12} : 6 πn + \frac{23 π}{4}

3 \cdot 2 πn + 3 \cdot \frac{23 π}{12}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

3 \cdot \frac{23 π}{12} = \frac{23 π}{4}

3 \cdot \frac{23 π}{12}

Multiply fractions:

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

= \frac{23 π 3}{12}

Multiply the numbers:

23 \cdot 3 = 69

= \frac{69 π}{12}

Cancel the common factor:

3

= \frac{23 π}{4}

= 6 πn + \frac{23 π}{4}

θ = 6 πn + \frac{23 π}{4}

θ = 6 πn + \frac{23 π}{4}

θ = 6 πn + \frac{23 π}{4}

θ = 6 πn + \frac{7 π}{4}, θ = 6 πn + \frac{23 π}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(θ/3-pi/4)= 1/2 ?
The general solution for cos(θ/3-pi/4)= 1/2 is θ=6pin+(7pi)/4 ,θ=6pin+(23pi)/4