English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

solvefor x,sin(x)-sin(2x)=0

Solution

solve for

x, sin (x) - sin (2 x) = 0

Solution

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

Degrees

x = 6 0^{\circ} + 24 0^{\circ} n, x = 18 0^{\circ} + 24 0^{\circ} n, x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

Solution steps

sin (x) - sin (2 x) = 0

Rewrite using trig identities

- sin (2 x) + sin (x)

Use the Sum to Product identity:

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

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

Simplify

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

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

\frac{x - 2 x}{2} = - \frac{x}{2}

\frac{x - 2 x}{2}

Add similar elements:

x - 2 x = - x

= \frac{- x}{2}

Apply the fraction rule:

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

= - \frac{x}{2}

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

Use the negative angle identity:

sin (- x) = - sin (x)

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

Remove parentheses:

(- a) = - a

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

Add similar elements:

x + 2 x = 3 x

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

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

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

Solving each part separately

cos (\frac{3 x}{2}) = 0 or sin (\frac{x}{2}) = 0

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

cos (\frac{3 x}{2}) = 0

General solutions for

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

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

Solve

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

\frac{3 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{3 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

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

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

Divide both sides by

3

3 x = π + 4 πn

Divide both sides by

3

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

Simplify

x = \frac{π}{3} + \frac{4 πn}{3}

x = \frac{π}{3} + \frac{4 πn}{3}

Solve

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

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

Divide both sides by

3

3 x = 3 π + 4 πn

Divide both sides by

3

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

Simplify

x = π + \frac{4 πn}{3}

x = π + \frac{4 πn}{3}

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

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

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

General solutions for

sin (\frac{x}{2}) = 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{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

\frac{x}{2} = 0 + 2 πn, \frac{x}{2} = π + 2 πn

Solve

\frac{x}{2} = 0 + 2 πn : x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{x}{2} = 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 πn

Simplify

x = 4 πn

x = 4 πn

Solve

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

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

Simplify

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

Combine all the solutions

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

Graph

Popular Examples

sin(θ)= 4/5 ,cos(θ)

sin (θ) = \frac{4}{5}, cos (θ)

sin(6x)=cos(3x)

sin (6 x) = cos (3 x)

sin(θ/2)-cos(θ/2)=0

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

sin(x)=(-3)/5

sin (x) = \frac{- 3}{5}

3cos^2(a)+sin^2(a)=2

3 cos^{2} (a) + sin^{2} (a) = 2