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

The general solution for 0=sin(θ/2)-sin(θ) is θ=(2pi)/3+(8pin)/3 ,θ=2pi+(8pin)/3 ,θ=8pin,θ=4pi+8pin

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Popular Trigonometry >

0=sin(θ/2)-sin(θ)

Solution

0 = sin (\frac{θ}{2}) - sin (θ)

Solution

θ = \frac{2 π}{3} + \frac{8 πn}{3}, θ = 2 π + \frac{8 πn}{3}, θ = 8 πn, θ = 4 π + 8 πn

Degrees

θ = 12 0^{\circ} + 48 0^{\circ} n, θ = 36 0^{\circ} + 48 0^{\circ} n, θ = 0^{\circ} + 144 0^{\circ} n, θ = 72 0^{\circ} + 144 0^{\circ} n

Solution steps

0 = sin (\frac{θ}{2}) - sin (θ)

Switch sides

sin (\frac{θ}{2}) - sin (θ) = 0

Rewrite using trig identities

sin (\frac{θ}{2}) - sin (θ)

Use the Sum to Product identity:

sin (s) - sin (t) = 2 sin (\frac{s - t}{2}) cos (\frac{s + t}{2})

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

Simplify

2 sin (\frac{\frac{θ}{2} - θ}{2}) cos (\frac{\frac{θ}{2} + θ}{2}) : - 2 cos (\frac{3 θ}{4}) sin (\frac{θ}{4})

2 sin (\frac{\frac{θ}{2} - θ}{2}) cos (\frac{\frac{θ}{2} + θ}{2})

\frac{\frac{θ}{2} - θ}{2} = - \frac{θ}{4}

\frac{\frac{θ}{2} - θ}{2}

Join

\frac{θ}{2} - θ : - \frac{θ}{2}

\frac{θ}{2} - θ

Convert element to fraction:

θ = \frac{θ 2}{2}

= \frac{θ}{2} - \frac{θ \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{θ - θ \cdot 2}{2}

Add similar elements:

θ - 2 θ = - θ

= \frac{- θ}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{θ}{2}

= \frac{- \frac{θ}{2}}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{θ}{2}}{2}

Apply the fraction rule:

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

\frac{\frac{θ}{2}}{2} = \frac{θ}{2 \cdot 2}

= - \frac{θ}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= - \frac{θ}{4}

= 2 sin (- \frac{θ}{4}) cos (\frac{\frac{θ}{2} + θ}{2})

Use the negative angle identity:

sin (- x) = - sin (x)

= 2 cos (\frac{\frac{θ}{2} + θ}{2}) (- sin (\frac{θ}{4}))

Remove parentheses:

(- a) = - a

= - 2 cos (\frac{\frac{θ}{2} + θ}{2}) sin (\frac{θ}{4})

\frac{\frac{θ}{2} + θ}{2} = \frac{3 θ}{4}

\frac{\frac{θ}{2} + θ}{2}

Join

\frac{θ}{2} + θ : \frac{3 θ}{2}

\frac{θ}{2} + θ

Convert element to fraction:

θ = \frac{θ 2}{2}

= \frac{θ}{2} + \frac{θ \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{θ + θ \cdot 2}{2}

Add similar elements:

θ + 2 θ = 3 θ

= \frac{3 θ}{2}

= \frac{\frac{3 θ}{2}}{2}

Apply the fraction rule:

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

= \frac{3 θ}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 θ}{4}

= - 2 cos (\frac{3 θ}{4}) sin (\frac{θ}{4})

= - 2 cos (\frac{3 θ}{4}) sin (\frac{θ}{4})

- 2 cos (\frac{3 θ}{4}) sin (\frac{θ}{4}) = 0

Solving each part separately

cos (\frac{3 θ}{4}) = 0 or sin (\frac{θ}{4}) = 0

cos (\frac{3 θ}{4}) = 0 : θ = \frac{2 π}{3} + \frac{8 πn}{3}, θ = 2 π + \frac{8 πn}{3}

cos (\frac{3 θ}{4}) = 0

General solutions for

cos (\frac{3 θ}{4}) = 0

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

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

Solve

\frac{3 θ}{4} = \frac{π}{2} + 2 πn : θ = \frac{2 π}{3} + \frac{8 πn}{3}

\frac{3 θ}{4} = \frac{π}{2} + 2 πn

Multiply both sides by

4

\frac{3 θ}{4} = \frac{π}{2} + 2 πn

Multiply both sides by

4

\frac{4 \cdot 3 θ}{4} = 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

Simplify

\frac{4 \cdot 3 θ}{4} = 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

Simplify

\frac{4 \cdot 3 θ}{4} : 3 θ

\frac{4 \cdot 3 θ}{4}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{12 θ}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3 θ

Simplify

4 \cdot \frac{π}{2} + 4 \cdot 2 πn : 2 π + 8 πn

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

3 θ = 2 π + 8 πn

3 θ = 2 π + 8 πn

3 θ = 2 π + 8 πn

Divide both sides by

3

3 θ = 2 π + 8 πn

Divide both sides by

3

\frac{3 θ}{3} = \frac{2 π}{3} + \frac{8 πn}{3}

Simplify

θ = \frac{2 π}{3} + \frac{8 πn}{3}

θ = \frac{2 π}{3} + \frac{8 πn}{3}

Solve

\frac{3 θ}{4} = \frac{3 π}{2} + 2 πn : θ = 2 π + \frac{8 πn}{3}

\frac{3 θ}{4} = \frac{3 π}{2} + 2 πn

Multiply both sides by

4

\frac{3 θ}{4} = \frac{3 π}{2} + 2 πn

Multiply both sides by

4

\frac{4 \cdot 3 θ}{4} = 4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

Simplify

\frac{4 \cdot 3 θ}{4} = 4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

Simplify

\frac{4 \cdot 3 θ}{4} : 3 θ

\frac{4 \cdot 3 θ}{4}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{12 θ}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3 θ

Simplify

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn : 6 π + 8 πn

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 4}{2}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{12 π}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

3 θ = 6 π + 8 πn

3 θ = 6 π + 8 πn

3 θ = 6 π + 8 πn

Divide both sides by

3

3 θ = 6 π + 8 πn

Divide both sides by

3

\frac{3 θ}{3} = \frac{6 π}{3} + \frac{8 πn}{3}

Simplify

θ = 2 π + \frac{8 πn}{3}

θ = 2 π + \frac{8 πn}{3}

θ = \frac{2 π}{3} + \frac{8 πn}{3}, θ = 2 π + \frac{8 πn}{3}

sin (\frac{θ}{4}) = 0 : θ = 8 πn, θ = 4 π + 8 πn

sin (\frac{θ}{4}) = 0

General solutions for

sin (\frac{θ}{4}) = 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}

\frac{θ}{4} = 0 + 2 πn, \frac{θ}{4} = π + 2 πn

\frac{θ}{4} = 0 + 2 πn, \frac{θ}{4} = π + 2 πn

Solve

\frac{θ}{4} = 0 + 2 πn : θ = 8 πn

\frac{θ}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{θ}{4} = 2 πn

Multiply both sides by

4

\frac{θ}{4} = 2 πn

Multiply both sides by

4

\frac{4 θ}{4} = 4 \cdot 2 πn

Simplify

θ = 8 πn

θ = 8 πn

Solve

\frac{θ}{4} = π + 2 πn : θ = 4 π + 8 πn

\frac{θ}{4} = π + 2 πn

Multiply both sides by

4

\frac{θ}{4} = π + 2 πn

Multiply both sides by

4

\frac{4 θ}{4} = 4 π + 4 \cdot 2 πn

Simplify

θ = 4 π + 8 πn

θ = 4 π + 8 πn

θ = 8 πn, θ = 4 π + 8 πn

Combine all the solutions

θ = \frac{2 π}{3} + \frac{8 πn}{3}, θ = 2 π + \frac{8 πn}{3}, θ = 8 πn, θ = 4 π + 8 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 0=sin(θ/2)-sin(θ) ?
The general solution for 0=sin(θ/2)-sin(θ) is θ=(2pi)/3+(8pin)/3 ,θ=2pi+(8pin)/3 ,θ=8pin,θ=4pi+8pin