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Popular Trigonometry >

sin(2x-8)=sin(x-4)

Solution

sin (2 x - 8) = sin (x - 4)

Solution

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

Degrees

x = 289.18311 \dots^{\circ} + 24 0^{\circ} n, x = 409.18311 \dots^{\circ} + 24 0^{\circ} n, x = 229.18311 \dots^{\circ} + 72 0^{\circ} n, x = 589.18311 \dots^{\circ} + 72 0^{\circ} n

Solution steps

sin (2 x - 8) = sin (x - 4)

Subtract

sin (x - 4)

from both sides

sin (2 x - 8) - sin (x - 4) = 0

Rewrite using trig identities

- sin (- 4 + x) + sin (- 8 + 2 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{- 8 + 2 x - ( - 4 + x )}{2}) cos (\frac{- 8 + 2 x - 4 + x}{2})

Simplify

2 sin (\frac{- 8 + 2 x - ( - 4 + x )}{2}) cos (\frac{- 8 + 2 x - 4 + x}{2}) : 2 sin (\frac{x - 4}{2}) cos (\frac{3 x - 12}{2})

2 sin (\frac{- 8 + 2 x - ( - 4 + x )}{2}) cos (\frac{- 8 + 2 x - 4 + x}{2})

Expand

- 8 + 2 x - (- 4 + x) : x - 4

- 8 + 2 x - (- 4 + x)

- (- 4 + x) : 4 - x

- (- 4 + x)

Distribute parentheses

= - (- 4) - (x)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= 4 - x

= - 8 + 2 x + 4 - x

Simplify

- 8 + 2 x + 4 - x : x - 4

- 8 + 2 x + 4 - x

Group like terms

= 2 x - x - 8 + 4

Add similar elements:

2 x - x = x

= x - 8 + 4

Add/Subtract the numbers:

- 8 + 4 = - 4

= x - 4

= x - 4

= 2 sin (\frac{x - 4}{2}) cos (\frac{2 x + x - 8 - 4}{2})

- 8 + 2 x - 4 + x = 3 x - 12

- 8 + 2 x - 4 + x

Group like terms

= 2 x + x - 8 - 4

Add similar elements:

2 x + x = 3 x

= 3 x - 8 - 4

Subtract the numbers:

- 8 - 4 = - 12

= 3 x - 12

= 2 sin (\frac{x - 4}{2}) cos (\frac{3 x - 12}{2})

= 2 sin (\frac{x - 4}{2}) cos (\frac{3 x - 12}{2})

2 cos (\frac{- 12 + 3 x}{2}) sin (\frac{- 4 + x}{2}) = 0

Solving each part separately

cos (\frac{- 12 + 3 x}{2}) = 0 or sin (\frac{- 4 + x}{2}) = 0

cos (\frac{- 12 + 3 x}{2}) = 0 : x = \frac{π}{3} + 4 + \frac{4 πn}{3}, x = π + 4 + \frac{4 πn}{3}

cos (\frac{- 12 + 3 x}{2}) = 0

General solutions for

cos (\frac{- 12 + 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{- 12 + 3 x}{2} = \frac{π}{2} + 2 πn, \frac{- 12 + 3 x}{2} = \frac{3 π}{2} + 2 πn

\frac{- 12 + 3 x}{2} = \frac{π}{2} + 2 πn, \frac{- 12 + 3 x}{2} = \frac{3 π}{2} + 2 πn

Solve

\frac{- 12 + 3 x}{2} = \frac{π}{2} + 2 πn : x = \frac{π}{3} + 4 + \frac{4 πn}{3}

\frac{- 12 + 3 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{- 12 + 3 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( - 12 + 3 x )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 12 + 3 x )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 12 + 3 x )}{2} : - 12 + 3 x

\frac{2 ( - 12 + 3 x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 12 + 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

- 12 + 3 x = π + 4 πn

- 12 + 3 x = π + 4 πn

- 12 + 3 x = π + 4 πn

Move

12

to the right side

- 12 + 3 x = π + 4 πn

Add

12

to both sides

- 12 + 3 x + 12 = π + 4 πn + 12

Simplify

3 x = π + 4 πn + 12

3 x = π + 4 πn + 12

Divide both sides by

3

3 x = π + 4 πn + 12

Divide both sides by

3

\frac{3 x}{3} = \frac{π}{3} + \frac{4 πn}{3} + \frac{12}{3}

Simplify

\frac{3 x}{3} = \frac{π}{3} + \frac{4 πn}{3} + \frac{12}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{π}{3} + \frac{4 πn}{3} + \frac{12}{3} : \frac{π}{3} + 4 + \frac{4 πn}{3}

\frac{π}{3} + \frac{4 πn}{3} + \frac{12}{3}

Group like terms

= \frac{π}{3} + \frac{12}{3} + \frac{4 πn}{3}

Divide the numbers:

\frac{12}{3} = 4

= \frac{π}{3} + 4 + \frac{4 πn}{3}

x = \frac{π}{3} + 4 + \frac{4 πn}{3}

x = \frac{π}{3} + 4 + \frac{4 πn}{3}

x = \frac{π}{3} + 4 + \frac{4 πn}{3}

Solve

\frac{- 12 + 3 x}{2} = \frac{3 π}{2} + 2 πn : x = π + 4 + \frac{4 πn}{3}

\frac{- 12 + 3 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{- 12 + 3 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( - 12 + 3 x )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 12 + 3 x )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( - 12 + 3 x )}{2} : - 12 + 3 x

\frac{2 ( - 12 + 3 x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 12 + 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

- 12 + 3 x = 3 π + 4 πn

- 12 + 3 x = 3 π + 4 πn

- 12 + 3 x = 3 π + 4 πn

Move

12

to the right side

- 12 + 3 x = 3 π + 4 πn

Add

12

to both sides

- 12 + 3 x + 12 = 3 π + 4 πn + 12

Simplify

3 x = 3 π + 4 πn + 12

3 x = 3 π + 4 πn + 12

Divide both sides by

3

3 x = 3 π + 4 πn + 12

Divide both sides by

3

\frac{3 x}{3} = \frac{3 π}{3} + \frac{4 πn}{3} + \frac{12}{3}

Simplify

\frac{3 x}{3} = \frac{3 π}{3} + \frac{4 πn}{3} + \frac{12}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{3 π}{3} + \frac{4 πn}{3} + \frac{12}{3} : π + 4 + \frac{4 πn}{3}

\frac{3 π}{3} + \frac{4 πn}{3} + \frac{12}{3}

Group like terms

= \frac{3 π}{3} + \frac{12}{3} + \frac{4 πn}{3}

Divide the numbers:

\frac{3}{3} = 1

= π + \frac{12}{3} + \frac{4 πn}{3}

Divide the numbers:

\frac{12}{3} = 4

= π + 4 + \frac{4 πn}{3}

x = π + 4 + \frac{4 πn}{3}

x = π + 4 + \frac{4 πn}{3}

x = π + 4 + \frac{4 πn}{3}

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

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

sin (\frac{- 4 + x}{2}) = 0

General solutions for

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

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

Solve

\frac{- 4 + x}{2} = 0 + 2 πn : x = 4 πn + 4

\frac{- 4 + x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{- 4 + x}{2} = 2 πn

Multiply both sides by

2

\frac{- 4 + x}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( - 4 + x )}{2} = 2 \cdot 2 πn

Simplify

- 4 + x = 4 πn

- 4 + x = 4 πn

Move

4

to the right side

- 4 + x = 4 πn

Add

4

to both sides

- 4 + x + 4 = 4 πn + 4

Simplify

x = 4 πn + 4

x = 4 πn + 4

Solve

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

\frac{- 4 + x}{2} = π + 2 πn

Multiply both sides by

2

\frac{- 4 + x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( - 4 + x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

- 4 + x = 2 π + 4 πn

- 4 + x = 2 π + 4 πn

Move

4

to the right side

- 4 + x = 2 π + 4 πn

Add

4

to both sides

- 4 + x + 4 = 2 π + 4 πn + 4

Simplify

x = 2 π + 4 πn + 4

x = 2 π + 4 πn + 4

x = 4 πn + 4, x = 2 π + 4 πn + 4

Combine all the solutions

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

Graph

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