What is the general solution for cos(x/3)-cos(x/(15))=0 ?

The general solution for cos(x/3)-cos(x/(15))=0 is x=15pin,x=(15pi)/2+15pin,x=10pin,x=5pi+10pin

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cos(x/3)-cos(x/(15))=0

Solution

cos (\frac{x}{3}) - cos (\frac{x}{15}) = 0

Solution

x = 15 πn, x = \frac{15 π}{2} + 15 πn, x = 10 πn, x = 5 π + 10 πn

Degrees

x = 0^{\circ} + 270 0^{\circ} n, x = 135 0^{\circ} + 270 0^{\circ} n, x = 0^{\circ} + 180 0^{\circ} n, x = 90 0^{\circ} + 180 0^{\circ} n

Solution steps

cos (\frac{x}{3}) - cos (\frac{x}{15}) = 0

Rewrite using trig identities

- cos (\frac{x}{15}) + cos (\frac{x}{3})

Use the Sum to Product identity:

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

= - 2 sin (\frac{\frac{x}{3} + \frac{x}{15}}{2}) sin (\frac{\frac{x}{3} - \frac{x}{15}}{2})

Simplify

- 2 sin (\frac{\frac{x}{3} + \frac{x}{15}}{2}) sin (\frac{\frac{x}{3} - \frac{x}{15}}{2}) : - 2 sin (\frac{x}{5}) sin (\frac{2 x}{15})

- 2 sin (\frac{\frac{x}{3} + \frac{x}{15}}{2}) sin (\frac{\frac{x}{3} - \frac{x}{15}}{2})

\frac{\frac{x}{3} + \frac{x}{15}}{2} = \frac{x}{5}

\frac{\frac{x}{3} + \frac{x}{15}}{2}

Join

\frac{x}{3} + \frac{x}{15} : \frac{2 x}{5}

\frac{x}{3} + \frac{x}{15}

Least Common Multiplier of

3, 15 : 15

3, 15

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

15 : 3 \cdot 5

15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 5

Multiply each factor the greatest number of times it occurs in either

3

15

= 3 \cdot 5

Multiply the numbers:

3 \cdot 5 = 15

= 15

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

15

For

\frac{x}{3} :

multiply the denominator and numerator by

5

\frac{x}{3} = \frac{x \cdot 5}{3 \cdot 5} = \frac{x \cdot 5}{15}

= \frac{x \cdot 5}{15} + \frac{x}{15}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 5 + x}{15}

Add similar elements:

5 x + x = 6 x

= \frac{6 x}{15}

Cancel the common factor:

3

= \frac{2 x}{5}

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

Apply the fraction rule:

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

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

Multiply the numbers:

5 \cdot 2 = 10

= \frac{2 x}{10}

Cancel the common factor:

2

= \frac{x}{5}

= - 2 sin (\frac{x}{5}) sin (\frac{\frac{x}{3} - \frac{x}{15}}{2})

\frac{\frac{x}{3} - \frac{x}{15}}{2} = \frac{2 x}{15}

\frac{\frac{x}{3} - \frac{x}{15}}{2}

Join

\frac{x}{3} - \frac{x}{15} : \frac{4 x}{15}

\frac{x}{3} - \frac{x}{15}

Least Common Multiplier of

3, 15 : 15

3, 15

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

15 : 3 \cdot 5

15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 5

Multiply each factor the greatest number of times it occurs in either

3

15

= 3 \cdot 5

Multiply the numbers:

3 \cdot 5 = 15

= 15

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

15

For

\frac{x}{3} :

multiply the denominator and numerator by

5

\frac{x}{3} = \frac{x \cdot 5}{3 \cdot 5} = \frac{x \cdot 5}{15}

= \frac{x \cdot 5}{15} - \frac{x}{15}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 5 - x}{15}

Add similar elements:

5 x - x = 4 x

= \frac{4 x}{15}

= \frac{\frac{4 x}{15}}{2}

Apply the fraction rule:

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

= \frac{4 x}{15 \cdot 2}

Multiply the numbers:

15 \cdot 2 = 30

= \frac{4 x}{30}

Cancel the common factor:

2

= \frac{2 x}{15}

= - 2 sin (\frac{x}{5}) sin (\frac{2 x}{15})

= - 2 sin (\frac{x}{5}) sin (\frac{2 x}{15})

- 2 sin (\frac{2 x}{15}) sin (\frac{x}{5}) = 0

Solving each part separately

sin (\frac{2 x}{15}) = 0 or sin (\frac{x}{5}) = 0

sin (\frac{2 x}{15}) = 0 : x = 15 πn, x = \frac{15 π}{2} + 15 πn

sin (\frac{2 x}{15}) = 0

General solutions for

sin (\frac{2 x}{15}) = 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{2 x}{15} = 0 + 2 πn, \frac{2 x}{15} = π + 2 πn

\frac{2 x}{15} = 0 + 2 πn, \frac{2 x}{15} = π + 2 πn

Solve

\frac{2 x}{15} = 0 + 2 πn : x = 15 πn

\frac{2 x}{15} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{2 x}{15} = 2 πn

Multiply both sides by

15

\frac{2 x}{15} = 2 πn

Multiply both sides by

15

\frac{15 \cdot 2 x}{15} = 15 \cdot 2 πn

Simplify

2 x = 30 πn

2 x = 30 πn

Divide both sides by

2

2 x = 30 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{30 πn}{2}

Simplify

x = 15 πn

x = 15 πn

Solve

\frac{2 x}{15} = π + 2 πn : x = \frac{15 π}{2} + 15 πn

\frac{2 x}{15} = π + 2 πn

Multiply both sides by

15

\frac{2 x}{15} = π + 2 πn

Multiply both sides by

15

\frac{15 \cdot 2 x}{15} = 15 π + 15 \cdot 2 πn

Simplify

2 x = 15 π + 30 πn

2 x = 15 π + 30 πn

Divide both sides by

2

2 x = 15 π + 30 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{15 π}{2} + \frac{30 πn}{2}

Simplify

x = \frac{15 π}{2} + 15 πn

x = \frac{15 π}{2} + 15 πn

x = 15 πn, x = \frac{15 π}{2} + 15 πn

sin (\frac{x}{5}) = 0 : x = 10 πn, x = 5 π + 10 πn

sin (\frac{x}{5}) = 0

General solutions for

sin (\frac{x}{5}) = 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}{5} = 0 + 2 πn, \frac{x}{5} = π + 2 πn

\frac{x}{5} = 0 + 2 πn, \frac{x}{5} = π + 2 πn

Solve

\frac{x}{5} = 0 + 2 πn : x = 10 πn

\frac{x}{5} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{5} = 2 πn

Multiply both sides by

5

\frac{x}{5} = 2 πn

Multiply both sides by

5

\frac{5 x}{5} = 5 \cdot 2 πn

Simplify

x = 10 πn

x = 10 πn

Solve

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

\frac{x}{5} = π + 2 πn

Multiply both sides by

5

\frac{x}{5} = π + 2 πn

Multiply both sides by

5

\frac{5 x}{5} = 5 π + 5 \cdot 2 πn

Simplify

x = 5 π + 10 πn

x = 5 π + 10 πn

x = 10 πn, x = 5 π + 10 πn

Combine all the solutions

x = 15 πn, x = \frac{15 π}{2} + 15 πn, x = 10 πn, x = 5 π + 10 πn

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Frequently Asked Questions (FAQ)

What is the general solution for cos(x/3)-cos(x/(15))=0 ?
The general solution for cos(x/3)-cos(x/(15))=0 is x=15pin,x=(15pi)/2+15pin,x=10pin,x=5pi+10pin