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

The general solution for cos(piz)=0 is z= 1/2+2n,z= 3/2+2n

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Popular Trigonometry >

cos(piz)=0

Solution

cos (π z) = 0

Solution

z = \frac{1}{2} + 2 n, z = \frac{3}{2} + 2 n

Degrees

z = 28.64788 \dots^{\circ} + 114.59155 \dots^{\circ} n, z = 85.94366 \dots^{\circ} + 114.59155 \dots^{\circ} n

Solution steps

cos (π z) = 0

General solutions for

cos (π z) = 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}

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

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

Solve

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

π z = \frac{π}{2} + 2 πn

Divide both sides by

π

π z = \frac{π}{2} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π z}{π} : z

\frac{π z}{π}

Cancel the common factor:

π

= z

Simplify

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

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

\frac{\frac{π}{2}}{π} = \frac{1}{2}

\frac{\frac{π}{2}}{π}

Apply the fraction rule:

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

= \frac{π}{2 π}

Cancel the common factor:

π

= \frac{1}{2}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= \frac{1}{2} + 2 n

z = \frac{1}{2} + 2 n

z = \frac{1}{2} + 2 n

z = \frac{1}{2} + 2 n

Solve

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

π z = \frac{3 π}{2} + 2 πn

Divide both sides by

π

π z = \frac{3 π}{2} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π z}{π} : z

\frac{π z}{π}

Cancel the common factor:

π

= z

Simplify

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

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

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

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

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

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

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

z = \frac{1}{2} + 2 n, z = \frac{3}{2} + 2 n

Graph

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Frequently Asked Questions (FAQ)

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