What is the general solution for sin(θ/2)= 1/2 sin(θ) ?

The general solution for sin(θ/2)= 1/2 sin(θ) is θ=4pin,θ=2pi+4pin

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sin(θ/2)= 1/2 sin(θ)

Solution

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

Solution

θ = 4 πn, θ = 2 π + 4 πn

Degrees

θ = 0^{\circ} + 72 0^{\circ} n, θ = 36 0^{\circ} + 72 0^{\circ} n

Solution steps

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

Subtract

\frac{1}{2} sin (θ)

from both sides

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

Simplify

sin (\frac{θ}{2}) - \frac{1}{2} sin (θ) : \frac{2 sin ( \frac{θ}{2} ) - sin ( θ )}{2}

sin (\frac{θ}{2}) - \frac{1}{2} sin (θ)

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

\frac{1}{2} sin (θ)

Multiply fractions:

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

= \frac{1 \cdot sin ( θ )}{2}

Multiply:

1 \cdot sin (θ) = sin (θ)

= \frac{sin ( θ )}{2}

= sin (\frac{θ}{2}) - \frac{sin ( θ )}{2}

Convert element to fraction:

sin (\frac{θ}{2}) = \frac{sin ( \frac{θ}{2} ) 2}{2}

= \frac{sin ( \frac{θ}{2} ) \cdot 2}{2} - \frac{sin ( θ )}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( \frac{θ}{2} ) \cdot 2 - sin ( θ )}{2}

\frac{2 sin ( \frac{θ}{2} ) - sin ( θ )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 sin (\frac{θ}{2}) - sin (θ) = 0

Let:

u = \frac{θ}{2}

2 sin (u) - sin (2 u) = 0

Rewrite using trig identities

- sin (2 u) + 2 sin (u)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= - 2 sin (u) cos (u) + 2 sin (u)

2 sin (u) - 2 cos (u) sin (u) = 0

Factor

2 sin (u) - 2 cos (u) sin (u) : - 2 sin (u) (cos (u) - 1)

2 sin (u) - 2 cos (u) sin (u)

Factor out common term

- 2 sin (u)

= - 2 sin (u) (- 1 + cos (u))

- 2 sin (u) (cos (u) - 1) = 0

Solving each part separately

sin (u) = 0 or cos (u) - 1 = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

General solutions for

sin (u) = 0

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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

Solve

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

cos (u) - 1 = 0 : u = 2 πn

cos (u) - 1 = 0

Move

1

to the right side

cos (u) - 1 = 0

Add

1

to both sides

cos (u) - 1 + 1 = 0 + 1

Simplify

cos (u) = 1

cos (u) = 1

General solutions for

cos (u) = 1

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}

u = 0 + 2 πn

u = 0 + 2 πn

Solve

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn

Substitute back

u = \frac{θ}{2}

\frac{θ}{2} = 2 πn : θ = 4 πn

\frac{θ}{2} = 2 πn

Multiply both sides by

2

\frac{θ}{2} = 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 \cdot 2 πn

Simplify

θ = 4 πn

θ = 4 πn

\frac{θ}{2} = π + 2 πn : θ = 2 π + 4 πn

\frac{θ}{2} = π + 2 πn

Multiply both sides by

2

\frac{θ}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 θ}{2} = 2 π + 2 \cdot 2 πn

Simplify

θ = 2 π + 4 πn

θ = 2 π + 4 πn

θ = 4 πn, θ = 2 π + 4 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(θ/2)= 1/2 sin(θ) ?
The general solution for sin(θ/2)= 1/2 sin(θ) is θ=4pin,θ=2pi+4pin