What is the general solution for cos(pix)=0.5 ?

The general solution for cos(pix)=0.5 is x= 1/3+2n,x= 5/3+2n

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cos(pix)=0.5

Solution

cos (π x) = 0.5

Solution

x = \frac{1}{3} + 2 n, x = \frac{5}{3} + 2 n

Degrees

x = 19.09859 \dots^{\circ} + 114.59155 \dots^{\circ} n, x = 95.49296 \dots^{\circ} + 114.59155 \dots^{\circ} n

Solution steps

cos (π x) = 0.5

General solutions for

cos (π x) = 0.5

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}{3} + 2 πn, π x = \frac{5 π}{3} + 2 πn

π x = \frac{π}{3} + 2 πn, π x = \frac{5 π}{3} + 2 πn

Solve

π x = \frac{π}{3} + 2 πn : x = \frac{1}{3} + 2 n

π x = \frac{π}{3} + 2 πn

Divide both sides by

π

π x = \frac{π}{3} + 2 πn

Divide both sides by

π

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

Simplify

\frac{π x}{π} = \frac{\frac{π}{3}}{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}{3}}{π} + \frac{2 πn}{π} : \frac{1}{3} + 2 n

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

\frac{\frac{π}{3}}{π} = \frac{1}{3}

\frac{\frac{π}{3}}{π}

Apply the fraction rule:

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

= \frac{π}{3 π}

Cancel the common factor:

π

= \frac{1}{3}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

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

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

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

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

Solve

π x = \frac{5 π}{3} + 2 πn : x = \frac{5}{3} + 2 n

π x = \frac{5 π}{3} + 2 πn

Divide both sides by

π

π x = \frac{5 π}{3} + 2 πn

Divide both sides by

π

\frac{π x}{π} = \frac{\frac{5 π}{3}}{π} + \frac{2 πn}{π}

Simplify

\frac{π x}{π} = \frac{\frac{5 π}{3}}{π} + \frac{2 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{5 π}{3}}{π} + \frac{2 πn}{π} : \frac{5}{3} + 2 n

\frac{\frac{5 π}{3}}{π} + \frac{2 πn}{π}

\frac{\frac{5 π}{3}}{π} = \frac{5}{3}

\frac{\frac{5 π}{3}}{π}

Apply the fraction rule:

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

= \frac{5 π}{3 π}

Cancel the common factor:

π

= \frac{5}{3}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= \frac{5}{3} + 2 n

x = \frac{5}{3} + 2 n

x = \frac{5}{3} + 2 n

x = \frac{5}{3} + 2 n

x = \frac{1}{3} + 2 n, x = \frac{5}{3} + 2 n

Popular Examples

cos(x)cot^2(x)-3cos(x)=0

cos (x) cot^{2} (x) - 3 cos (x) = 0

-3cos^2(θ)-6cos(θ)+3=-cos(θ)+5

- 3 cos^{2} (θ) - 6 cos (θ) + 3 = - cos (θ) + 5

cos(x)-1=cos(2x)

cos (x) - 1 = cos (2 x)

sin(x+pi/4)=(sqrt(2))/2

sin (x + \frac{π}{4}) = \frac{2}{2}

tan(θ)=2.4

tan (θ) = 2.4

Frequently Asked Questions (FAQ)

What is the general solution for cos(pix)=0.5 ?
The general solution for cos(pix)=0.5 is x= 1/3+2n,x= 5/3+2n