What is the general solution for cos(5x)-cos(x)=sin(5x)-sin(x) ?

The general solution for cos(5x)-cos(x)=sin(5x)-sin(x) is x=pin,x= pi/2+pin,x= pi/4+(pin)/3

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Popular Trigonometry >

cos(5x)-cos(x)=sin(5x)-sin(x)

Solution

cos (5 x) - cos (x) = sin (5 x) - sin (x)

Solution

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

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 6 0^{\circ} n

Solution steps

cos (5 x) - cos (x) = sin (5 x) - sin (x)

Subtract

sin (5 x) - sin (x)

from both sides

cos (5 x) - cos (x) - sin (5 x) + sin (x) = 0

Rewrite using trig identities

cos (5 x) - cos (x) - sin (5 x) + sin (x)

Use the Sum to Product identity:

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

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

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

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

\frac{x - 5 x}{2} = - 2 x

\frac{x - 5 x}{2}

Add similar elements:

x - 5 x = - 4 x

= \frac{- 4 x}{2}

Apply the fraction rule:

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

= - \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= - 2 x

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

Use the negative angle identity:

sin (- x) = - sin (x)

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

Remove parentheses:

(- a) = - a

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

\frac{x + 5 x}{2} = 3 x

\frac{x + 5 x}{2}

Add similar elements:

x + 5 x = 6 x

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

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

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

2 sin (\frac{5 x + x}{2}) sin (\frac{5 x - x}{2}) = 2 sin (3 x) sin (2 x)

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

\frac{5 x + x}{2} = 3 x

\frac{5 x + x}{2}

Add similar elements:

5 x + x = 6 x

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

= 2 sin (3 x) sin (\frac{5 x - x}{2})

\frac{5 x - x}{2} = 2 x

\frac{5 x - x}{2}

Add similar elements:

5 x - x = 4 x

= \frac{4 x}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 x

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

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

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

Factor

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

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

Rewrite as

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

Factor out common term

2 sin (2 x)

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

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

Solving each part separately

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

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

sin (2 x) = 0

General solutions for

sin (2 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}

2 x = 0 + 2 πn, 2 x = π + 2 πn

2 x = 0 + 2 πn, 2 x = π + 2 πn

Solve

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

Divide both sides by

2

2 x = 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplify

x = πn

x = πn

Solve

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Divide both sides by

2

2 x = π + 2 πn

Divide both sides by

2

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

Simplify

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

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

Graph

Popular Examples

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tan(x)= 7/10

cos(5x)+cos(x)=2cos(2x)

cos(x)= 22/24

sin(x)=(1.2)/(1.6)

Frequently Asked Questions (FAQ)

What is the general solution for cos(5x)-cos(x)=sin(5x)-sin(x) ?
The general solution for cos(5x)-cos(x)=sin(5x)-sin(x) is x=pin,x= pi/2+pin,x= pi/4+(pin)/3