What is the general solution for sin(4x)=sin(3x) ?

The general solution for sin(4x)=sin(3x) is x= pi/7+(4pin)/7 ,x=(3pi)/7+(4pin)/7 ,x=4pin,x=2pi+4pin

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Popular Trigonometry >

sin(4x)=sin(3x)

Solution

sin (4 x) = sin (3 x)

Solution

x = \frac{π}{7} + \frac{4 πn}{7}, x = \frac{3 π}{7} + \frac{4 πn}{7}, x = 4 πn, x = 2 π + 4 πn

Degrees

x = 25.71428 \dots^{\circ} + 102.85714 \dots^{\circ} n, x = 77.14285 \dots^{\circ} + 102.85714 \dots^{\circ} n, x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

Solution steps

sin (4 x) = sin (3 x)

Subtract

sin (3 x)

from both sides

sin (4 x) - sin (3 x) = 0

Rewrite using trig identities

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

Simplify

2 sin (\frac{4 x - 3 x}{2}) cos (\frac{4 x + 3 x}{2}) : 2 sin (\frac{x}{2}) cos (\frac{7 x}{2})

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

Add similar elements:

4 x - 3 x = x

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

Add similar elements:

4 x + 3 x = 7 x

= 2 sin (\frac{x}{2}) cos (\frac{7 x}{2})

= 2 sin (\frac{x}{2}) cos (\frac{7 x}{2})

2 cos (\frac{7 x}{2}) sin (\frac{x}{2}) = 0

Solving each part separately

cos (\frac{7 x}{2}) = 0 or sin (\frac{x}{2}) = 0

cos (\frac{7 x}{2}) = 0 : x = \frac{π}{7} + \frac{4 πn}{7}, x = \frac{3 π}{7} + \frac{4 πn}{7}

cos (\frac{7 x}{2}) = 0

General solutions for

cos (\frac{7 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{7 x}{2} = \frac{π}{2} + 2 πn, \frac{7 x}{2} = \frac{3 π}{2} + 2 πn

\frac{7 x}{2} = \frac{π}{2} + 2 πn, \frac{7 x}{2} = \frac{3 π}{2} + 2 πn

Solve

\frac{7 x}{2} = \frac{π}{2} + 2 πn : x = \frac{π}{7} + \frac{4 πn}{7}

\frac{7 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{7 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{2 \cdot 7 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 7 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 7 x}{2} : 7 x

\frac{2 \cdot 7 x}{2}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{14 x}{2}

Divide the numbers:

\frac{14}{2} = 7

= 7 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

7 x = π + 4 πn

7 x = π + 4 πn

7 x = π + 4 πn

Divide both sides by

7

7 x = π + 4 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{π}{7} + \frac{4 πn}{7}

Simplify

x = \frac{π}{7} + \frac{4 πn}{7}

x = \frac{π}{7} + \frac{4 πn}{7}

Solve

\frac{7 x}{2} = \frac{3 π}{2} + 2 πn : x = \frac{3 π}{7} + \frac{4 πn}{7}

\frac{7 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{7 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{2 \cdot 7 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 7 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplify

\frac{2 \cdot 7 x}{2} : 7 x

\frac{2 \cdot 7 x}{2}

Multiply the numbers:

2 \cdot 7 = 14

= \frac{14 x}{2}

Divide the numbers:

\frac{14}{2} = 7

= 7 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

7 x = 3 π + 4 πn

7 x = 3 π + 4 πn

7 x = 3 π + 4 πn

Divide both sides by

7

7 x = 3 π + 4 πn

Divide both sides by

7

\frac{7 x}{7} = \frac{3 π}{7} + \frac{4 πn}{7}

Simplify

x = \frac{3 π}{7} + \frac{4 πn}{7}

x = \frac{3 π}{7} + \frac{4 πn}{7}

x = \frac{π}{7} + \frac{4 πn}{7}, x = \frac{3 π}{7} + \frac{4 πn}{7}

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}{7} + \frac{4 πn}{7}, x = \frac{3 π}{7} + \frac{4 πn}{7}, x = 4 πn, x = 2 π + 4 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(4x)=sin(3x) ?
The general solution for sin(4x)=sin(3x) is x= pi/7+(4pin)/7 ,x=(3pi)/7+(4pin)/7 ,x=4pin,x=2pi+4pin