What is the general solution for sin(4x)-cos(4x)=sin(2x)-cos(2x) ?

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

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Popular >

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

Solution

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

Solution

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

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 6 0^{\circ} n

Solution steps

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

Subtract

sin (2 x) - cos (2 x)

from both sides

sin (4 x) - cos (4 x) - sin (2 x) + cos (2 x) = 0

Rewrite using trig identities

cos (2 x) - cos (4 x) - sin (2 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})

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

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

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

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

\frac{4 x - 2 x}{2}

Add similar elements:

4 x - 2 x = 2 x

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

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

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

\frac{4 x + 2 x}{2}

Add similar elements:

4 x + 2 x = 6 x

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

= 2 sin (x) cos (3 x)

= cos (2 x) - cos (4 x) + 2 sin (x) cos (3 x)

Use the Sum to Product identity:

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

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

Simplify

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

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

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

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

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

\frac{2 x + 4 x}{2}

Add similar elements:

2 x + 4 x = 6 x

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

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

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

\frac{2 x - 4 x}{2}

Add similar elements:

2 x - 4 x = - 2 x

= \frac{- 2 x}{2}

Apply the fraction rule:

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

= - \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= - x

= 2 sin (3 x) sin (- x)

Use the negative angle identity:

sin (- x) = - sin (x)

= 2 (- sin (x)) sin (3 x)

= 2 cos (3 x) sin (x) - (- 2 sin (x) sin (3 x))

Apply rule

- (- a) = a

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

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

2 cos (3 x) sin (x) + 2 sin (3 x) sin (x) = 0

Factor

2 cos (3 x) sin (x) + 2 sin (3 x) sin (x) : 2 sin (x) (cos (3 x) + sin (3 x))

2 cos (3 x) sin (x) + 2 sin (3 x) sin (x)

Rewrite as

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

Factor out common term

2 sin (x)

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

2 sin (x) (cos (3 x) + sin (3 x)) = 0

Solving each part separately

sin (x) = 0 or cos (3 x) + sin (3 x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

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

cos (3 x) + sin (3 x) = 0

Rewrite using trig identities

cos (3 x) + sin (3 x) = 0

Divide both sides by

cos (3 x), cos (3 x) \neq = 0

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

Simplify

1 + \frac{sin ( 3 x )}{cos ( 3 x )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

1 + tan (3 x) = 0

1 + tan (3 x) = 0

Move

1

to the right side

1 + tan (3 x) = 0

Subtract

1

from both sides

1 + tan (3 x) - 1 = 0 - 1

Simplify

tan (3 x) = - 1

tan (3 x) = - 1

General solutions for

tan (3 x) = - 1

tan (x)

periodicity table with

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

3 x = \frac{3 π}{4} + πn

3 x = \frac{3 π}{4} + πn

Solve

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

3 x = \frac{3 π}{4} + πn

Divide both sides by

3

3 x = \frac{3 π}{4} + πn

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

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

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

\frac{\frac{3 π}{4}}{3} = \frac{π}{4}

\frac{\frac{3 π}{4}}{3}

Apply the fraction rule:

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

= \frac{3 π}{4 \cdot 3}

Multiply the numbers:

4 \cdot 3 = 12

= \frac{3 π}{12}

Cancel the common factor:

3

= \frac{π}{4}

= \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

x = \frac{π}{4} + \frac{πn}{3}

Combine all the solutions

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

Graph

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Frequently Asked Questions (FAQ)

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