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

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

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cos(pi/3 x-pi/4)=(sqrt(2))/2

Solution

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

Solution

x = \frac{3}{2} + 6 n, x = 6 n + 6

Degrees

x = 85.94366 \dots^{\circ} + 343.77467 \dots^{\circ} n, x = 343.77467 \dots^{\circ} + 343.77467 \dots^{\circ} n

Solution steps

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

General solutions for

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

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

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Add

\frac{π}{4}

to both sides

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

Simplify

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

Simplify

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

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

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= \frac{π}{3} x

Simplify

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

\frac{π}{4} + 2 πn + \frac{π}{4}

Group like terms

= \frac{π}{4} + \frac{π}{4} + 2 πn

Combine the fractions

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

Apply rule

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

= \frac{π + π}{4}

Add similar elements:

π + π = 2 π

= \frac{2 π}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{π}{2} + 2 πn

\frac{π}{3} x = \frac{π}{2} + 2 πn

\frac{π}{3} x = \frac{π}{2} + 2 πn

\frac{π}{3} x = \frac{π}{2} + 2 πn

Multiply both sides by

3

\frac{π}{3} x = \frac{π}{2} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

3 \cdot \frac{π}{3} x : π x

3 \cdot \frac{π}{3} x

Multiply fractions:

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

= \frac{3 π}{3} x

Cancel the common factor:

3

= x π

Simplify

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

3 \cdot \frac{π}{2} + 3 \cdot 2 πn

Multiply

3 \cdot \frac{π}{2} : \frac{3 π}{2}

3 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 3}{2}

= \frac{3 π}{2} + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= \frac{3 π}{2} + 6 πn

π x = \frac{3 π}{2} + 6 πn

π x = \frac{3 π}{2} + 6 πn

π x = \frac{3 π}{2} + 6 πn

Divide both sides by

π

π x = \frac{3 π}{2} + 6 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

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

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

\frac{\frac{3 π}{2}}{π} = \frac{3}{2}

\frac{\frac{3 π}{2}}{π}

Apply the fraction rule:

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

= \frac{3 π}{2 π}

Cancel the common factor:

π

= \frac{3}{2}

\frac{6 πn}{π} = 6 n

\frac{6 πn}{π}

Cancel the common factor:

π

= 6 n

= \frac{3}{2} + 6 n

x = \frac{3}{2} + 6 n

x = \frac{3}{2} + 6 n

x = \frac{3}{2} + 6 n

Solve

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

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

Move

\frac{π}{4}

to the right side

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

Add

\frac{π}{4}

to both sides

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

Simplify

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

Simplify

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

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

Add similar elements:

- \frac{π}{4} + \frac{π}{4} = 0

= \frac{π}{3} x

Simplify

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

\frac{7 π}{4} + 2 πn + \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{4} + \frac{7 π}{4}

Combine the fractions

\frac{π}{4} + \frac{7 π}{4} : 2 π

Apply rule

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

= \frac{π + 7 π}{4}

Add similar elements:

π + 7 π = 8 π

= \frac{8 π}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2 π

= 2 πn + 2 π

\frac{π}{3} x = 2 πn + 2 π

\frac{π}{3} x = 2 πn + 2 π

\frac{π}{3} x = 2 πn + 2 π

Multiply both sides by

3

\frac{π}{3} x = 2 πn + 2 π

Multiply both sides by

3

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

Simplify

π x = 6 πn + 6 π

π x = 6 πn + 6 π

Divide both sides by

π

π x = 6 πn + 6 π

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{6 πn}{π} + \frac{6 π}{π} : 6 n + 6

\frac{6 πn}{π} + \frac{6 π}{π}

Cancel

\frac{6 πn}{π} : 6 n

\frac{6 πn}{π}

Cancel the common factor:

π

= 6 n

= 6 n + \frac{6 π}{π}

Cancel

\frac{6 π}{π} : 6

\frac{6 π}{π}

Cancel the common factor:

π

= 6

= 6 n + 6

x = 6 n + 6

x = 6 n + 6

x = 6 n + 6

x = \frac{3}{2} + 6 n, x = 6 n + 6

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Frequently Asked Questions (FAQ)

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