What is the general solution for 0.5=cos(pi/6 x) ?

The general solution for 0.5=cos(pi/6 x) is x=2+12n,x=10+12n

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0.5=cos(pi/6 x)

Solution

0.5 = cos (\frac{π}{6} x)

Solution

x = 2 + 12 n, x = 10 + 12 n

Degrees

x = 114.59155 \dots^{\circ} + 687.54935 \dots^{\circ} n, x = 572.95779 \dots^{\circ} + 687.54935 \dots^{\circ} n

Solution steps

0.5 = cos (\frac{π}{6} x)

Switch sides

cos (\frac{π}{6} x) = 0.5

General solutions for

cos (\frac{π}{6} 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}

\frac{π}{6} x = \frac{π}{3} + 2 πn, \frac{π}{6} x = \frac{5 π}{3} + 2 πn

\frac{π}{6} x = \frac{π}{3} + 2 πn, \frac{π}{6} x = \frac{5 π}{3} + 2 πn

Solve

\frac{π}{6} x = \frac{π}{3} + 2 πn : x = 2 + 12 n

\frac{π}{6} x = \frac{π}{3} + 2 πn

Multiply both sides by

6

\frac{π}{6} x = \frac{π}{3} + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x = 6 \cdot \frac{π}{3} + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x = 6 \cdot \frac{π}{3} + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiply fractions:

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

= \frac{6 π}{6} x

Cancel the common factor:

6

= x π

Simplify

6 \cdot \frac{π}{3} + 6 \cdot 2 πn : 2 π + 12 πn

6 \cdot \frac{π}{3} + 6 \cdot 2 πn

6 \cdot \frac{π}{3} = 2 π

6 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 6}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 2 π + 12 πn

π x = 2 π + 12 πn

π x = 2 π + 12 πn

π x = 2 π + 12 πn

Divide both sides by

π

π x = 2 π + 12 πn

Divide both sides by

π

\frac{π x}{π} = \frac{2 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} = \frac{2 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{2 π}{π} + \frac{12 πn}{π} : 2 + 12 n

\frac{2 π}{π} + \frac{12 πn}{π}

Cancel

\frac{2 π}{π} : 2

\frac{2 π}{π}

Cancel the common factor:

π

= 2

= 2 + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 2 + 12 n

x = 2 + 12 n

x = 2 + 12 n

x = 2 + 12 n

Solve

\frac{π}{6} x = \frac{5 π}{3} + 2 πn : x = 10 + 12 n

\frac{π}{6} x = \frac{5 π}{3} + 2 πn

Multiply both sides by

6

\frac{π}{6} x = \frac{5 π}{3} + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x = 6 \cdot \frac{5 π}{3} + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x = 6 \cdot \frac{5 π}{3} + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiply fractions:

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

= \frac{6 π}{6} x

Cancel the common factor:

6

= x π

Simplify

6 \cdot \frac{5 π}{3} + 6 \cdot 2 πn : 10 π + 12 πn

6 \cdot \frac{5 π}{3} + 6 \cdot 2 πn

6 \cdot \frac{5 π}{3} = 10 π

6 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 6}{3}

Multiply the numbers:

5 \cdot 6 = 30

= \frac{30 π}{3}

Divide the numbers:

\frac{30}{3} = 10

= 10 π

6 \cdot 2 πn = 12 πn

6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 12 πn

= 10 π + 12 πn

π x = 10 π + 12 πn

π x = 10 π + 12 πn

π x = 10 π + 12 πn

Divide both sides by

π

π x = 10 π + 12 πn

Divide both sides by

π

\frac{π x}{π} = \frac{10 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} = \frac{10 π}{π} + \frac{12 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{10 π}{π} + \frac{12 πn}{π} : 10 + 12 n

\frac{10 π}{π} + \frac{12 πn}{π}

Cancel

\frac{10 π}{π} : 10

\frac{10 π}{π}

Cancel the common factor:

π

= 10

= 10 + \frac{12 πn}{π}

Cancel

\frac{12 πn}{π} : 12 n

\frac{12 πn}{π}

Cancel the common factor:

π

= 12 n

= 10 + 12 n

x = 10 + 12 n

x = 10 + 12 n

x = 10 + 12 n

x = 2 + 12 n, x = 10 + 12 n

Popular Examples

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tan (x) = \frac{sec ( x )}{cos ( x )}

sin(x)=-0.707

sin (x) = - 0.707

2sin^2(x)+5sin(x)=0

2 sin^{2} (x) + 5 sin (x) = 0

Frequently Asked Questions (FAQ)

What is the general solution for 0.5=cos(pi/6 x) ?
The general solution for 0.5=cos(pi/6 x) is x=2+12n,x=10+12n