What is the general solution for cos(pix)=0,0<= x<= 2pi ?

Q: What is the general solution for cos(pix)=0,0<= x<= 2pi ?

The general solution for cos(pix)=0,0<= x<= 2pi is x= 1/2 ,x= 3/2 ,x= 5/2 ,x= 7/2 ,x= 9/2 ,x= 11/2

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cos(pix)=0,0<= x<= 2pi

Solution

cos (π x) = 0, 0 \leq x \leq 2 π

Solution

x = \frac{1}{2}, x = \frac{3}{2}, x = \frac{5}{2}, x = \frac{7}{2}, x = \frac{9}{2}, x = \frac{11}{2}

Degrees

x = 28.64788 \dots^{\circ}, x = 85.94366 \dots^{\circ}, x = 143.23944 \dots^{\circ}, x = 200.53522 \dots^{\circ}, x = 257.83100 \dots^{\circ}, x = 315.12678 \dots^{\circ}

Solution steps

cos (π x) = 0, 0 \leq x \leq 2 π

General solutions for

cos (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}

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

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

Solve

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

π x = \frac{π}{2} + 2 πn

Divide both sides by

π

π x = \frac{π}{2} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

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

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

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

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

Solve

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

π x = \frac{3 π}{2} + 2 πn

Divide both sides by

π

π x = \frac{3 π}{2} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

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

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

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

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

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

Solutions for the range

0 \leq x \leq 2 π

x = \frac{1}{2}, x = \frac{3}{2}, x = \frac{5}{2}, x = \frac{7}{2}, x = \frac{9}{2}, x = \frac{11}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(pix)=0,0<= x<= 2pi ?
The general solution for cos(pix)=0,0<= x<= 2pi is x= 1/2 ,x= 3/2 ,x= 5/2 ,x= 7/2 ,x= 9/2 ,x= 11/2