What is the general solution for 4cos(pi/4 (x+3))=0 ?

The general solution for 4cos(pi/4 (x+3))=0 is x=8n-1,x=8n+3

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4cos(pi/4 (x+3))=0

Solution

4 cos (\frac{π}{4} (x + 3)) = 0

Solution

x = 8 n - 1, x = 8 n + 3

Degrees

x = - 57.29577 \dots^{\circ} + 458.36623 \dots^{\circ} n, x = 171.88733 \dots^{\circ} + 458.36623 \dots^{\circ} n

Solution steps

4 cos (\frac{π}{4} (x + 3)) = 0

Divide both sides by

4

4 cos (\frac{π}{4} (x + 3)) = 0

Divide both sides by

4

\frac{4 cos ( \frac{π}{4} ( x + 3 ) )}{4} = \frac{0}{4}

Simplify

cos (\frac{π}{4} (x + 3)) = 0

cos (\frac{π}{4} (x + 3)) = 0

General solutions for

cos (\frac{π}{4} (x + 3)) = 0

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

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

Solve

\frac{π}{4} (x + 3) = \frac{π}{2} + 2 πn : x = 8 n - 1

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

4 \cdot \frac{π}{4} (x + 3) : π (x + 3)

4 \cdot \frac{π}{4} (x + 3)

Multiply fractions:

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

= \frac{4 π}{4} (x + 3)

Cancel the common factor:

4

= (x + 3) π

Simplify

4 \cdot \frac{π}{2} + 4 \cdot 2 πn : 2 π + 8 πn

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

π (x + 3) = 2 π + 8 πn

π (x + 3) = 2 π + 8 πn

π (x + 3) = 2 π + 8 πn

Divide both sides by

π

π (x + 3) = 2 π + 8 πn

Divide both sides by

π

\frac{π ( x + 3 )}{π} = \frac{2 π}{π} + \frac{8 πn}{π}

Simplify

\frac{π ( x + 3 )}{π} = \frac{2 π}{π} + \frac{8 πn}{π}

Simplify

\frac{π ( x + 3 )}{π} : x + 3

\frac{π ( x + 3 )}{π}

Cancel the common factor:

π

= x + 3

Simplify

\frac{2 π}{π} + \frac{8 πn}{π} : 2 + 8 n

\frac{2 π}{π} + \frac{8 πn}{π}

Cancel

\frac{2 π}{π} : 2

\frac{2 π}{π}

Cancel the common factor:

π

= 2

= 2 + \frac{8 πn}{π}

Cancel

\frac{8 πn}{π} : 8 n

\frac{8 πn}{π}

Cancel the common factor:

π

= 8 n

= 2 + 8 n

x + 3 = 2 + 8 n

x + 3 = 2 + 8 n

x + 3 = 2 + 8 n

Move

3

to the right side

x + 3 = 2 + 8 n

Subtract

3

from both sides

x + 3 - 3 = 2 + 8 n - 3

Simplify

x = 8 n - 1

x = 8 n - 1

Solve

\frac{π}{4} (x + 3) = \frac{3 π}{2} + 2 πn : x = 8 n + 3

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

4 \cdot \frac{π}{4} (x + 3) : π (x + 3)

4 \cdot \frac{π}{4} (x + 3)

Multiply fractions:

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

= \frac{4 π}{4} (x + 3)

Cancel the common factor:

4

= (x + 3) π

Simplify

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn : 6 π + 8 πn

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 4}{2}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{12 π}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

π (x + 3) = 6 π + 8 πn

π (x + 3) = 6 π + 8 πn

π (x + 3) = 6 π + 8 πn

Divide both sides by

π

π (x + 3) = 6 π + 8 πn

Divide both sides by

π

\frac{π ( x + 3 )}{π} = \frac{6 π}{π} + \frac{8 πn}{π}

Simplify

\frac{π ( x + 3 )}{π} = \frac{6 π}{π} + \frac{8 πn}{π}

Simplify

\frac{π ( x + 3 )}{π} : x + 3

\frac{π ( x + 3 )}{π}

Cancel the common factor:

π

= x + 3

Simplify

\frac{6 π}{π} + \frac{8 πn}{π} : 6 + 8 n

\frac{6 π}{π} + \frac{8 πn}{π}

Cancel

\frac{6 π}{π} : 6

\frac{6 π}{π}

Cancel the common factor:

π

= 6

= 6 + \frac{8 πn}{π}

Cancel

\frac{8 πn}{π} : 8 n

\frac{8 πn}{π}

Cancel the common factor:

π

= 8 n

= 6 + 8 n

x + 3 = 6 + 8 n

x + 3 = 6 + 8 n

x + 3 = 6 + 8 n

Move

3

to the right side

x + 3 = 6 + 8 n

Subtract

3

from both sides

x + 3 - 3 = 6 + 8 n - 3

Simplify

x = 8 n + 3

x = 8 n + 3

x = 8 n - 1, x = 8 n + 3

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Frequently Asked Questions (FAQ)

What is the general solution for 4cos(pi/4 (x+3))=0 ?
The general solution for 4cos(pi/4 (x+3))=0 is x=8n-1,x=8n+3