What is the general solution for cos(2pix)=0 ?

The general solution for cos(2pix)=0 is x= 1/4+n,x= 3/4+n

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cos(2pix)=0

Solution

cos (2 π x) = 0

Solution

x = \frac{1}{4} + n, x = \frac{3}{4} + n

Degrees

x = 14.32394 \dots^{\circ}, x = 42.97183 \dots^{\circ}

Solution steps

cos (2 π x) = 0

General solutions for

cos (2 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}

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

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

Solve

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

2 π x = \frac{π}{2} + 2 πn

Divide both sides by

2 π

2 π x = \frac{π}{2} + 2 πn

Divide both sides by

2 π

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

Simplify

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

Simplify

\frac{2 π x}{2 π} : x

\frac{2 π x}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{π}{2}}{2 π} + \frac{2 πn}{2 π} : \frac{1}{4} + n

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

\frac{\frac{π}{2}}{2 π} = \frac{1}{4}

\frac{\frac{π}{2}}{2 π}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2 π}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4 π}

Cancel the common factor:

π

= \frac{1}{4}

\frac{2 πn}{2 π} = n

\frac{2 πn}{2 π}

Cancel

\frac{2 πn}{2 π} : n

\frac{2 πn}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{πn}{π}

Cancel the common factor:

π

= n

= n

= \frac{1}{4} + n

x = \frac{1}{4} + n

x = \frac{1}{4} + n

x = \frac{1}{4} + n

Solve

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

2 π x = \frac{3 π}{2} + 2 πn

Divide both sides by

2 π

2 π x = \frac{3 π}{2} + 2 πn

Divide both sides by

2 π

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

Simplify

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

Simplify

\frac{2 π x}{2 π} : x

\frac{2 π x}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{3 π}{2}}{2 π} + \frac{2 πn}{2 π} : \frac{3}{4} + n

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

\frac{\frac{3 π}{2}}{2 π} = \frac{3}{4}

\frac{\frac{3 π}{2}}{2 π}

Apply the fraction rule:

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

= \frac{3 π}{2 \cdot 2 π}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 π}{4 π}

Cancel the common factor:

π

= \frac{3}{4}

\frac{2 πn}{2 π} = n

\frac{2 πn}{2 π}

Cancel

\frac{2 πn}{2 π} : n

\frac{2 πn}{2 π}

Divide the numbers:

\frac{2}{2} = 1

= \frac{πn}{π}

Cancel the common factor:

π

= n

= n

= \frac{3}{4} + n

x = \frac{3}{4} + n

x = \frac{3}{4} + n

x = \frac{3}{4} + n

x = \frac{1}{4} + n, x = \frac{3}{4} + n

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Frequently Asked Questions (FAQ)

What is the general solution for cos(2pix)=0 ?
The general solution for cos(2pix)=0 is x= 1/4+n,x= 3/4+n