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

The general solution for cos(pi/x)=0 is x= 2/(1+4n),x= 2/(3+4n);\quad n\ne-1/4 ,n\ne-3/4

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cos(pi/x)=0

Solution

cos (\frac{π}{x}) = 0

Solution

x = \frac{2}{1 + 4 n}, x = \frac{2}{3 + 4 n}; n \neq = - \frac{1}{4}, n \neq = - \frac{3}{4}

Degrees

x = 0^{\circ} + 22.91831 \dots^{\circ} n, x = 0^{\circ} + 16.37022 \dots^{\circ} n

Solution steps

cos (\frac{π}{x}) = 0

General solutions for

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

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

Solve

\frac{π}{x} = \frac{π}{2} + 2 πn : x = \frac{2}{1 + 4 n}; n \neq = - \frac{1}{4}

\frac{π}{x} = \frac{π}{2} + 2 πn

Multiply both sides by

x

\frac{π}{x} = \frac{π}{2} + 2 πn

Multiply both sides by

x

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

Simplify

π = \frac{π}{2} x + 2 πn x

π = \frac{π}{2} x + 2 πn x

Switch sides

\frac{π}{2} x + 2 πn x = π

Multiply both sides by

2

\frac{π}{2} x + 2 πn x = π

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{π}{2} x \cdot 2 : π x

\frac{π}{2} x \cdot 2

Multiply fractions:

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

= \frac{2 π}{2} x

Cancel the common factor:

2

= x π

Simplify

2 πn x \cdot 2 : 4 πn x

2 πn x \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn x

Simplify

π 2 : 2 π

π 2

Apply the commutative law:

π 2 = 2 π

2 π

π x + 4 πn x = 2 π

π x + 4 πn x = 2 π

π x + 4 πn x = 2 π

Factor

π x + 4 πn x : π x (1 + 4 n)

π x + 4 πn x

Factor out common term

x π

= x π (1 + 4 n)

π x (1 + 4 n) = 2 π

Divide both sides by

π (1 + 4 n); n \neq = - \frac{1}{4}

π x (1 + 4 n) = 2 π

Divide both sides by

π (1 + 4 n); n \neq = - \frac{1}{4}

\frac{π x ( 1 + 4 n )}{π ( 1 + 4 n )} = \frac{2 π}{π ( 1 + 4 n )}; n \neq = - \frac{1}{4}

Simplify

x = \frac{2}{1 + 4 n}; n \neq = - \frac{1}{4}

x = \frac{2}{1 + 4 n}; n \neq = - \frac{1}{4}

Solve

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

\frac{π}{x} = \frac{3 π}{2} + 2 πn

Multiply both sides by

x

\frac{π}{x} = \frac{3 π}{2} + 2 πn

Multiply both sides by

x

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

Simplify

π = \frac{3 π}{2} x + 2 πn x

π = \frac{3 π}{2} x + 2 πn x

Switch sides

\frac{3 π}{2} x + 2 πn x = π

Multiply both sides by

2

\frac{3 π}{2} x + 2 πn x = π

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{3 π}{2} x \cdot 2 : 3 π x

\frac{3 π}{2} x \cdot 2

Multiply fractions:

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

= \frac{3 \cdot 2 π}{2} x

Cancel the common factor:

2

= x \cdot 3 π

Simplify

2 πn x \cdot 2 : 4 πn x

2 πn x \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn x

Simplify

π 2 : 2 π

π 2

Apply the commutative law:

π 2 = 2 π

2 π

3 π x + 4 πn x = 2 π

3 π x + 4 πn x = 2 π

3 π x + 4 πn x = 2 π

Factor

3 π x + 4 πn x : π x (3 + 4 n)

3 π x + 4 πn x

Factor out common term

x π

= x π (3 + 4 n)

π x (3 + 4 n) = 2 π

Divide both sides by

π (3 + 4 n); n \neq = - \frac{3}{4}

π x (3 + 4 n) = 2 π

Divide both sides by

π (3 + 4 n); n \neq = - \frac{3}{4}

\frac{π x ( 3 + 4 n )}{π ( 3 + 4 n )} = \frac{2 π}{π ( 3 + 4 n )}; n \neq = - \frac{3}{4}

Simplify

x = \frac{2}{3 + 4 n}; n \neq = - \frac{3}{4}

x = \frac{2}{3 + 4 n}; n \neq = - \frac{3}{4}

x = \frac{2}{1 + 4 n}, x = \frac{2}{3 + 4 n}; n \neq = - \frac{1}{4}, n \neq = - \frac{3}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(pi/x)=0 ?
The general solution for cos(pi/x)=0 is x= 2/(1+4n),x= 2/(3+4n);\quad n\ne-1/4 ,n\ne-3/4