What is the general solution for 2cos(1/2 x)-1=0 ?

The general solution for 2cos(1/2 x)-1=0 is x=(2pi)/3+4pin,x=(10pi)/3+4pin

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2cos(1/2 x)-1=0

Solution

2 cos (\frac{1}{2} x) - 1 = 0

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

2 cos (\frac{1}{2} x) - 1 = 0

Move

1

to the right side

2 cos (\frac{1}{2} x) - 1 = 0

Add

1

to both sides

2 cos (\frac{1}{2} x) - 1 + 1 = 0 + 1

Simplify

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

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

General solutions for

cos (\frac{1}{2} x) = \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{1}{2} x = \frac{π}{3} + 2 πn, \frac{1}{2} x = \frac{5 π}{3} + 2 πn

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

Solve

\frac{1}{2} x = \frac{π}{3} + 2 πn : x = \frac{2 π}{3} + 4 πn

\frac{1}{2} x = \frac{π}{3} + 2 πn

Multiply both sides by

2

\frac{1}{2} x = \frac{π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

= \frac{1 \cdot 2}{2} x

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= 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{1}{2} x = \frac{5 π}{3} + 2 πn : x = \frac{10 π}{3} + 4 πn

\frac{1}{2} x = \frac{5 π}{3} + 2 πn

Multiply both sides by

2

\frac{1}{2} x = \frac{5 π}{3} + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x = 2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x = 2 \cdot \frac{5 π}{3} + 2 \cdot 2 πn

Simplify

2 \cdot \frac{1}{2} x : x

2 \cdot \frac{1}{2} x

Multiply fractions:

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

= \frac{1 \cdot 2}{2} x

Cancel the common factor:

2

= x \cdot 1

Multiply:

x \cdot 1 = x

= 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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2cos(1/2 x)-1=0 ?
The general solution for 2cos(1/2 x)-1=0 is x=(2pi)/3+4pin,x=(10pi)/3+4pin