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Popular >

sin(3x)=sin(2x+50)

Solution

sin (3 x) = sin (2 x + 50)

Solution

x = \frac{π}{5} - 10 + \frac{4 πn}{5}, x = \frac{3 π}{5} - 10 + \frac{4 πn}{5}, x = 4 πn + 50, x = 2 π + 4 πn + 50

Degrees

x = - 536.95779 \dots^{\circ} + 14 4^{\circ} n, x = - 464.95779 \dots^{\circ} + 14 4^{\circ} n, x = 2864.78897 \dots^{\circ} + 72 0^{\circ} n, x = 3224.78897 \dots^{\circ} + 72 0^{\circ} n

Solution steps

sin (3 x) = sin (2 x + 50)

Subtract

sin (2 x + 50)

from both sides

sin (3 x) - sin (2 x + 50) = 0

Rewrite using trig identities

sin (3 x) - sin (50 + 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{3 x - ( 50 + 2 x )}{2}) cos (\frac{3 x + 50 + 2 x}{2})

Simplify

2 sin (\frac{3 x - ( 50 + 2 x )}{2}) cos (\frac{3 x + 50 + 2 x}{2}) : 2 sin (\frac{x - 50}{2}) cos (\frac{5 x + 50}{2})

2 sin (\frac{3 x - ( 50 + 2 x )}{2}) cos (\frac{3 x + 50 + 2 x}{2})

Expand

3 x - (50 + 2 x) : x - 50

3 x - (50 + 2 x)

- (50 + 2 x) : - 50 - 2 x

- (50 + 2 x)

Distribute parentheses

= - (50) - (2 x)

Apply minus-plus rules

+ (- a) = - a

= - 50 - 2 x

= 3 x - 50 - 2 x

Simplify

3 x - 50 - 2 x : x - 50

3 x - 50 - 2 x

Group like terms

= 3 x - 2 x - 50

Add similar elements:

3 x - 2 x = x

= x - 50

= x - 50

= 2 sin (\frac{x - 50}{2}) cos (\frac{3 x + 2 x + 50}{2})

3 x + 50 + 2 x = 5 x + 50

3 x + 50 + 2 x

Group like terms

= 3 x + 2 x + 50

Add similar elements:

3 x + 2 x = 5 x

= 5 x + 50

= 2 sin (\frac{x - 50}{2}) cos (\frac{5 x + 50}{2})

= 2 sin (\frac{x - 50}{2}) cos (\frac{5 x + 50}{2})

2 cos (\frac{50 + 5 x}{2}) sin (\frac{- 50 + x}{2}) = 0

Solving each part separately

cos (\frac{50 + 5 x}{2}) = 0 or sin (\frac{- 50 + x}{2}) = 0

cos (\frac{50 + 5 x}{2}) = 0 : x = \frac{π}{5} - 10 + \frac{4 πn}{5}, x = \frac{3 π}{5} - 10 + \frac{4 πn}{5}

cos (\frac{50 + 5 x}{2}) = 0

General solutions for

cos (\frac{50 + 5 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{50 + 5 x}{2} = \frac{π}{2} + 2 πn, \frac{50 + 5 x}{2} = \frac{3 π}{2} + 2 πn

\frac{50 + 5 x}{2} = \frac{π}{2} + 2 πn, \frac{50 + 5 x}{2} = \frac{3 π}{2} + 2 πn

Solve

\frac{50 + 5 x}{2} = \frac{π}{2} + 2 πn : x = \frac{π}{5} - 10 + \frac{4 πn}{5}

\frac{50 + 5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{50 + 5 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( 50 + 5 x )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 50 + 5 x )}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 50 + 5 x )}{2} : 50 + 5 x

\frac{2 ( 50 + 5 x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 50 + 5 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

50 + 5 x = π + 4 πn

50 + 5 x = π + 4 πn

50 + 5 x = π + 4 πn

Move

50

to the right side

50 + 5 x = π + 4 πn

Subtract

50

from both sides

50 + 5 x - 50 = π + 4 πn - 50

Simplify

5 x = π + 4 πn - 50

5 x = π + 4 πn - 50

Divide both sides by

5

5 x = π + 4 πn - 50

Divide both sides by

5

\frac{5 x}{5} = \frac{π}{5} + \frac{4 πn}{5} - \frac{50}{5}

Simplify

\frac{5 x}{5} = \frac{π}{5} + \frac{4 πn}{5} - \frac{50}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{π}{5} + \frac{4 πn}{5} - \frac{50}{5} : \frac{π}{5} - 10 + \frac{4 πn}{5}

\frac{π}{5} + \frac{4 πn}{5} - \frac{50}{5}

Group like terms

= \frac{π}{5} - \frac{50}{5} + \frac{4 πn}{5}

Divide the numbers:

\frac{50}{5} = 10

= \frac{π}{5} - 10 + \frac{4 πn}{5}

x = \frac{π}{5} - 10 + \frac{4 πn}{5}

x = \frac{π}{5} - 10 + \frac{4 πn}{5}

x = \frac{π}{5} - 10 + \frac{4 πn}{5}

Solve

\frac{50 + 5 x}{2} = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{5} - 10 + \frac{4 πn}{5}

\frac{50 + 5 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{50 + 5 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 ( 50 + 5 x )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 50 + 5 x )}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 ( 50 + 5 x )}{2} : 50 + 5 x

\frac{2 ( 50 + 5 x )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 50 + 5 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

50 + 5 x = 3 π + 4 πn

50 + 5 x = 3 π + 4 πn

50 + 5 x = 3 π + 4 πn

Move

50

to the right side

50 + 5 x = 3 π + 4 πn

Subtract

50

from both sides

50 + 5 x - 50 = 3 π + 4 πn - 50

Simplify

5 x = 3 π + 4 πn - 50

5 x = 3 π + 4 πn - 50

Divide both sides by

5

5 x = 3 π + 4 πn - 50

Divide both sides by

5

\frac{5 x}{5} = \frac{3 π}{5} + \frac{4 πn}{5} - \frac{50}{5}

Simplify

\frac{5 x}{5} = \frac{3 π}{5} + \frac{4 πn}{5} - \frac{50}{5}

Simplify

\frac{5 x}{5} : x

\frac{5 x}{5}

Divide the numbers:

\frac{5}{5} = 1

= x

Simplify

\frac{3 π}{5} + \frac{4 πn}{5} - \frac{50}{5} : \frac{3 π}{5} - 10 + \frac{4 πn}{5}

\frac{3 π}{5} + \frac{4 πn}{5} - \frac{50}{5}

Group like terms

= \frac{3 π}{5} - \frac{50}{5} + \frac{4 πn}{5}

Divide the numbers:

\frac{50}{5} = 10

= \frac{3 π}{5} - 10 + \frac{4 πn}{5}

x = \frac{3 π}{5} - 10 + \frac{4 πn}{5}

x = \frac{3 π}{5} - 10 + \frac{4 πn}{5}

x = \frac{3 π}{5} - 10 + \frac{4 πn}{5}

x = \frac{π}{5} - 10 + \frac{4 πn}{5}, x = \frac{3 π}{5} - 10 + \frac{4 πn}{5}

sin (\frac{- 50 + x}{2}) = 0 : x = 4 πn + 50, x = 2 π + 4 πn + 50

sin (\frac{- 50 + x}{2}) = 0

General solutions for

sin (\frac{- 50 + 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{- 50 + x}{2} = 0 + 2 πn, \frac{- 50 + x}{2} = π + 2 πn

\frac{- 50 + x}{2} = 0 + 2 πn, \frac{- 50 + x}{2} = π + 2 πn

Solve

\frac{- 50 + x}{2} = 0 + 2 πn : x = 4 πn + 50

\frac{- 50 + x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{- 50 + x}{2} = 2 πn

Multiply both sides by

2

\frac{- 50 + x}{2} = 2 πn

Multiply both sides by

2

\frac{2 ( - 50 + x )}{2} = 2 \cdot 2 πn

Simplify

- 50 + x = 4 πn

- 50 + x = 4 πn

Move

50

to the right side

- 50 + x = 4 πn

Add

50

to both sides

- 50 + x + 50 = 4 πn + 50

Simplify

x = 4 πn + 50

x = 4 πn + 50

Solve

\frac{- 50 + x}{2} = π + 2 πn : x = 2 π + 4 πn + 50

\frac{- 50 + x}{2} = π + 2 πn

Multiply both sides by

2

\frac{- 50 + x}{2} = π + 2 πn

Multiply both sides by

2

\frac{2 ( - 50 + x )}{2} = 2 π + 2 \cdot 2 πn

Simplify

- 50 + x = 2 π + 4 πn

- 50 + x = 2 π + 4 πn

Move

50

to the right side

- 50 + x = 2 π + 4 πn

Add

50

to both sides

- 50 + x + 50 = 2 π + 4 πn + 50

Simplify

x = 2 π + 4 πn + 50

x = 2 π + 4 πn + 50

x = 4 πn + 50, x = 2 π + 4 πn + 50

Combine all the solutions

x = \frac{π}{5} - 10 + \frac{4 πn}{5}, x = \frac{3 π}{5} - 10 + \frac{4 πn}{5}, x = 4 πn + 50, x = 2 π + 4 πn + 50

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