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

The general solution for cos(x/2)=1-cos(x/2) is x=(2pi)/3+4pin,x=(10pi)/3+4pin

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

Solution

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

Solution

x = \frac{2 π}{3} + 4 πn, x = \frac{10 π}{3} + 4 πn

Degrees

x = 12 0^{\circ} + 72 0^{\circ} n, x = 60 0^{\circ} + 72 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cos (\frac{x}{2}) = u

u = 1 - u

u = 1 - u : u = \frac{1}{2}

u = 1 - u

Move

u

to the left side

u = 1 - u

Add

u

to both sides

u + u = 1 - u + u

Simplify

2 u = 1

2 u = 1

Divide both sides by

2

2 u = 1

Divide both sides by

2

\frac{2 u}{2} = \frac{1}{2}

Simplify

u = \frac{1}{2}

u = \frac{1}{2}

Substitute back

u = cos (\frac{x}{2})

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

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

cos (\frac{x}{2}) = \frac{1}{2} : x = \frac{2 π}{3} + 4 πn, x = \frac{10 π}{3} + 4 πn

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

General solutions for

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

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

Solve

\frac{x}{2} = \frac{π}{3} + 2 πn : x = \frac{2 π}{3} + 4 πn

\frac{x}{2} = \frac{π}{3} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{3} + 2 \cdot 2 πn : \frac{2 π}{3} + 4 πn

2 \cdot \frac{π}{3} + 2 \cdot 2 πn

Multiply

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 2}{3}

= \frac{2 π}{3} + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

Solve

\frac{x}{2} = \frac{5 π}{3} + 2 πn : x = \frac{10 π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{3} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{5 π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn : \frac{10 π}{3} + 4 πn

2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{3} = \frac{10 π}{3}

2 \cdot \frac{5 π}{3}

Multiply fractions:

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

= \frac{5 π 2}{3}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{10 π}{3} + 4 πn

x = \frac{10 π}{3} + 4 πn

x = \frac{10 π}{3} + 4 πn

x = \frac{10 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn, x = \frac{10 π}{3} + 4 πn

Combine all the solutions

x = \frac{2 π}{3} + 4 πn, x = \frac{10 π}{3} + 4 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x/2)=1-cos(x/2) ?
The general solution for cos(x/2)=1-cos(x/2) is x=(2pi)/3+4pin,x=(10pi)/3+4pin