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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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

Popular Examples

sin(x)*cos(x)-3cos(x)=0

sin (x) \cdot cos (x) - 3 cos (x) = 0

9.8*sin(α)-0.2*9.8*cos(α)=0.47

9.8 \cdot sin (α) - 0.2 \cdot 9.8 \cdot cos (α) = 0.47

-sin(x)+3cos(x)=0

- sin (x) + 3 cos (x) = 0

tan^2(θ)=(sin^2(θ))/(cos(θ))

tan^{2} (θ) = \frac{sin ^{2} ( θ )}{cos ( θ )}

tan(x)=1,17

tan (x) = 1, 17

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