What is the general solution for cos((pi(x-7))/3)= 1/2 ?

The general solution for cos((pi(x-7))/3)= 1/2 is x=6n+8,x=6n+12

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cos((pi(x-7))/3)= 1/2

Solution

cos (\frac{π ( x - 7 )}{3}) = \frac{1}{2}

Solution

x = 6 n + 8, x = 6 n + 12

Degrees

x = 458.36623 \dots^{\circ} + 343.77467 \dots^{\circ} n, x = 687.54935 \dots^{\circ} + 343.77467 \dots^{\circ} n

Solution steps

cos (\frac{π ( x - 7 )}{3}) = \frac{1}{2}

General solutions for

cos (\frac{π ( x - 7 )}{3}) = \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{π ( x - 7 )}{3} = \frac{π}{3} + 2 πn, \frac{π ( x - 7 )}{3} = \frac{5 π}{3} + 2 πn

\frac{π ( x - 7 )}{3} = \frac{π}{3} + 2 πn, \frac{π ( x - 7 )}{3} = \frac{5 π}{3} + 2 πn

Solve

\frac{π ( x - 7 )}{3} = \frac{π}{3} + 2 πn : x = 6 n + 8

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

Multiply both sides by

3

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

Multiply both sides by

3

\frac{3 π ( x - 7 )}{3} = 3 \cdot \frac{π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 π ( x - 7 )}{3} = 3 \cdot \frac{π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 π ( x - 7 )}{3} : π (x - 7)

\frac{3 π ( x - 7 )}{3}

Divide the numbers:

\frac{3}{3} = 1

= π (x - 7)

Simplify

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

3 \cdot \frac{π}{3} + 3 \cdot 2 πn

3 \cdot \frac{π}{3} = π

3 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 3}{3}

Cancel the common factor:

3

= π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= π + 6 πn

π (x - 7) = π + 6 πn

π (x - 7) = π + 6 πn

π (x - 7) = π + 6 πn

Divide both sides by

π

π (x - 7) = π + 6 πn

Divide both sides by

π

\frac{π ( x - 7 )}{π} = \frac{π}{π} + \frac{6 πn}{π}

Simplify

\frac{π ( x - 7 )}{π} = \frac{π}{π} + \frac{6 πn}{π}

Simplify

\frac{π ( x - 7 )}{π} : x - 7

\frac{π ( x - 7 )}{π}

Cancel the common factor:

π

= x - 7

Simplify

\frac{π}{π} + \frac{6 πn}{π} : 1 + 6 n

\frac{π}{π} + \frac{6 πn}{π}

Apply rule

\frac{a}{a} = 1

\frac{π}{π} = 1

= 1 + \frac{6 πn}{π}

Cancel

\frac{6 πn}{π} : 6 n

\frac{6 πn}{π}

Cancel the common factor:

π

= 6 n

= 1 + 6 n

x - 7 = 1 + 6 n

x - 7 = 1 + 6 n

x - 7 = 1 + 6 n

Move

7

to the right side

x - 7 = 1 + 6 n

Add

7

to both sides

x - 7 + 7 = 1 + 6 n + 7

Simplify

x = 6 n + 8

x = 6 n + 8

Solve

\frac{π ( x - 7 )}{3} = \frac{5 π}{3} + 2 πn : x = 6 n + 12

\frac{π ( x - 7 )}{3} = \frac{5 π}{3} + 2 πn

Multiply both sides by

3

\frac{π ( x - 7 )}{3} = \frac{5 π}{3} + 2 πn

Multiply both sides by

3

\frac{3 π ( x - 7 )}{3} = 3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 π ( x - 7 )}{3} = 3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn

Simplify

\frac{3 π ( x - 7 )}{3} : π (x - 7)

\frac{3 π ( x - 7 )}{3}

Divide the numbers:

\frac{3}{3} = 1

= π (x - 7)

Simplify

3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn : 5 π + 6 πn

3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn

3 \cdot \frac{5 π}{3} = 5 π

3 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 3}{3}

Cancel the common factor:

3

= 5 π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= 5 π + 6 πn

π (x - 7) = 5 π + 6 πn

π (x - 7) = 5 π + 6 πn

π (x - 7) = 5 π + 6 πn

Divide both sides by

π

π (x - 7) = 5 π + 6 πn

Divide both sides by

π

\frac{π ( x - 7 )}{π} = \frac{5 π}{π} + \frac{6 πn}{π}

Simplify

\frac{π ( x - 7 )}{π} = \frac{5 π}{π} + \frac{6 πn}{π}

Simplify

\frac{π ( x - 7 )}{π} : x - 7

\frac{π ( x - 7 )}{π}

Cancel the common factor:

π

= x - 7

Simplify

\frac{5 π}{π} + \frac{6 πn}{π} : 5 + 6 n

\frac{5 π}{π} + \frac{6 πn}{π}

Cancel

\frac{5 π}{π} : 5

\frac{5 π}{π}

Cancel the common factor:

π

= 5

= 5 + \frac{6 πn}{π}

Cancel

\frac{6 πn}{π} : 6 n

\frac{6 πn}{π}

Cancel the common factor:

π

= 6 n

= 5 + 6 n

x - 7 = 5 + 6 n

x - 7 = 5 + 6 n

x - 7 = 5 + 6 n

Move

7

to the right side

x - 7 = 5 + 6 n

Add

7

to both sides

x - 7 + 7 = 5 + 6 n + 7

Simplify

x = 6 n + 12

x = 6 n + 12

x = 6 n + 8, x = 6 n + 12

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Frequently Asked Questions (FAQ)

What is the general solution for cos((pi(x-7))/3)= 1/2 ?
The general solution for cos((pi(x-7))/3)= 1/2 is x=6n+8,x=6n+12