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

The general solution for 2sin(x/2)-1=0 is x= pi/3+4pin,x=(5pi)/3+4pin

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2sin(x/2)-1=0

Solution

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

Solution

x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn

Degrees

x = 6 0^{\circ} + 72 0^{\circ} n, x = 30 0^{\circ} + 72 0^{\circ} n

Solution steps

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

Divide both sides by

2

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

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

Solve

\frac{x}{2} = \frac{π}{6} + 2 πn : x = \frac{π}{3} + 4 πn

\frac{x}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{6} + 2 πn

Multiply both sides by

2

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

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{6} + 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}{6} + 2 \cdot 2 πn : \frac{π}{3} + 4 πn

2 \cdot \frac{π}{6} + 2 \cdot 2 πn

2 \cdot \frac{π}{6} = \frac{π}{3}

2 \cdot \frac{π}{6}

Multiply fractions:

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

= \frac{π 2}{6}

Cancel the common factor:

2

= \frac{π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

x = \frac{π}{3} + 4 πn

Solve

\frac{x}{2} = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{3} + 4 πn

\frac{x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{5 π}{6} + 2 πn

Multiply both sides by

2

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

Simplify

\frac{2 x}{2} = 2 \cdot \frac{5 π}{6} + 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}{6} + 2 \cdot 2 πn : \frac{5 π}{3} + 4 πn

2 \cdot \frac{5 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{5 π}{3} + 4 πn

x = \frac{π}{3} + 4 πn, x = \frac{5 π}{3} + 4 πn

Popular Examples

cot(θ/7)=sqrt(12)

cot (\frac{θ}{7}) = 12

cos(x)sin(x)+sin(x)=0

cos (x) sin (x) + sin (x) = 0

2sin(θ)+6=5

2 sin (θ) + 6 = 5

3sin(θ)-sqrt(3)=sin(θ)

3 sin (θ) - 3 = sin (θ)

cos(x)= 15/17

cos (x) = \frac{15}{17}

Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x/2)-1=0 ?
The general solution for 2sin(x/2)-1=0 is x= pi/3+4pin,x=(5pi)/3+4pin