What is the general solution for 2cos(x/3)=1 ?

The general solution for 2cos(x/3)=1 is x=pi+6pin,x=5pi+6pin

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2cos(x/3)=1

Solution

2 cos (\frac{x}{3}) = 1

Solution

x = π + 6 πn, x = 5 π + 6 πn

Degrees

x = 18 0^{\circ} + 108 0^{\circ} n, x = 90 0^{\circ} + 108 0^{\circ} n

Solution steps

2 cos (\frac{x}{3}) = 1

Divide both sides by

2

2 cos (\frac{x}{3}) = 1

Divide both sides by

2

\frac{2 cos ( \frac{x}{3} )}{2} = \frac{1}{2}

Simplify

cos (\frac{x}{3}) = \frac{1}{2}

cos (\frac{x}{3}) = \frac{1}{2}

General solutions for

cos (\frac{x}{3}) = \frac{1}{2}

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

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

Solve

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

\frac{x}{3} = \frac{π}{3} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{π}{3} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

3 \cdot \frac{π}{3} + 3 \cdot 2 πn

3 \cdot \frac{π}{3} = π

3 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 3}{3}

Cancel the common factor:

3

= π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= π + 6 πn

x = π + 6 πn

x = π + 6 πn

x = π + 6 πn

Solve

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

\frac{x}{3} = \frac{5 π}{3} + 2 πn

Multiply both sides by

3

\frac{x}{3} = \frac{5 π}{3} + 2 πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

3 \cdot \frac{5 π}{3} + 3 \cdot 2 πn

3 \cdot \frac{5 π}{3} = 5 π

3 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 3}{3}

Cancel the common factor:

3

= 5 π

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 πn

= 5 π + 6 πn

x = 5 π + 6 πn

x = 5 π + 6 πn

x = 5 π + 6 πn

x = π + 6 πn, x = 5 π + 6 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos(x/3)=1 ?
The general solution for 2cos(x/3)=1 is x=pi+6pin,x=5pi+6pin