What is the general solution for solvefor x,sin(x+r/4)+sin(x-r/3)=0 ?

The general solution for solvefor x,sin(x+r/4)+sin(x-r/3)=0 is x=2pin+r/(24),x=pi+2pin+r/(24)

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solvefor x,sin(x+r/4)+sin(x-r/3)=0

Solution

solve for

x, sin (x + \frac{r}{4}) + sin (x - \frac{r}{3}) = 0

Solution

x = 2 πn + \frac{r}{24}, x = π + 2 πn + \frac{r}{24}

Solution steps

sin (x + \frac{r}{4}) + sin (x - \frac{r}{3}) = 0

Rewrite using trig identities

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

Use the Sum to Product identity:

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

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

Simplify

2 sin (\frac{x + \frac{r}{4} + x - \frac{r}{3}}{2}) cos (\frac{x + \frac{r}{4} - ( x - \frac{r}{3} )}{2}) : 2 cos (\frac{7 r}{24}) sin (\frac{24 x - r}{24})

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

\frac{x + \frac{r}{4} + x - \frac{r}{3}}{2} = \frac{24 x - r}{24}

\frac{x + \frac{r}{4} + x - \frac{r}{3}}{2}

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

x + \frac{r}{4} + x - \frac{r}{3}

Group like terms

= x + x + \frac{r}{4} - \frac{r}{3}

Add similar elements:

x + x = 2 x

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

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

Join

2 x + \frac{r}{4} - \frac{r}{3} : \frac{24 x - r}{12}

2 x + \frac{r}{4} - \frac{r}{3}

Convert element to fraction:

2 x = \frac{2 x}{1}

= \frac{2 x}{1} + \frac{r}{4} - \frac{r}{3}

Least Common Multiplier of

1, 4, 3 : 12

1, 4, 3

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Compute a number comprised of factors that appear in at least one of the following:

1, 4, 3

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

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

12

For

\frac{2 x}{1} :

multiply the denominator and numerator by

12

\frac{2 x}{1} = \frac{2 x \cdot 12}{1 \cdot 12} = \frac{24 x}{12}

For

\frac{r}{4} :

multiply the denominator and numerator by

3

\frac{r}{4} = \frac{r \cdot 3}{4 \cdot 3} = \frac{r \cdot 3}{12}

For

\frac{r}{3} :

multiply the denominator and numerator by

4

\frac{r}{3} = \frac{r \cdot 4}{3 \cdot 4} = \frac{r \cdot 4}{12}

= \frac{24 x}{12} + \frac{r \cdot 3}{12} - \frac{r \cdot 4}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{24 x + r \cdot 3 - r \cdot 4}{12}

Add similar elements:

3 r - 4 r = - r

= \frac{24 x - r}{12}

= \frac{\frac{24 x - r}{12}}{2}

Apply the fraction rule:

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

= \frac{24 x - r}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{24 x - r}{24}

= 2 sin (\frac{24 x - r}{24}) cos (\frac{x - ( x - \frac{r}{3} ) + \frac{r}{4}}{2})

\frac{x + \frac{r}{4} - ( x - \frac{r}{3} )}{2} = \frac{7 r}{24}

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

Join

x + \frac{r}{4} - (x - \frac{r}{3}) : \frac{7 r}{12}

x + \frac{r}{4} - (x - \frac{r}{3})

Convert element to fraction:

x = \frac{x 4}{4}, (x - \frac{r}{3}) = \frac{( x - \frac{r}{3} ) 4}{4}

= \frac{x \cdot 4}{4} + \frac{r}{4} - \frac{( x - \frac{r}{3} ) \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{x \cdot 4 + r - ( x - \frac{r}{3} ) \cdot 4}{4}

Expand

x \cdot 4 + r - (x - \frac{r}{3}) \cdot 4 : \frac{4 r}{3} + r

x \cdot 4 + r - (x - \frac{r}{3}) \cdot 4

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

Expand

- 4 (x - \frac{r}{3}) : - 4 x + \frac{4 r}{3}

- 4 (x - \frac{r}{3})

Apply the distributive law:

a (b - c) = ab - a c

a = - 4, b = x, c = \frac{r}{3}

= - 4 x - (- 4) \frac{r}{3}

Apply minus-plus rules

- (- a) = a

= - 4 x + 4 \cdot \frac{r}{3}

Multiply

4 \cdot \frac{r}{3} : \frac{4 r}{3}

4 \cdot \frac{r}{3}

Multiply fractions:

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

= \frac{r \cdot 4}{3}

= - 4 x + \frac{4 r}{3}

= x \cdot 4 + r - 4 x + \frac{4 r}{3}

Simplify

x \cdot 4 + r - 4 x + \frac{4 r}{3} : \frac{4 r}{3} + r

x \cdot 4 + r - 4 x + \frac{4 r}{3}

Group like terms

= 4 x - 4 x + \frac{4 r}{3} + r

Add similar elements:

4 x - 4 x = 0

= \frac{4 r}{3} + r

= \frac{4 r}{3} + r

= \frac{\frac{4 r}{3} + r}{4}

Join

\frac{4 r}{3} + r : \frac{7 r}{3}

\frac{4 r}{3} + r

Convert element to fraction:

r = \frac{r 3}{3}

= \frac{4 r}{3} + \frac{r \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{4 r + r \cdot 3}{3}

Add similar elements:

4 r + 3 r = 7 r

= \frac{7 r}{3}

= \frac{\frac{7 r}{3}}{4}

Apply the fraction rule:

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

= \frac{7 r}{3 \cdot 4}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{7 r}{12}

= \frac{\frac{7 r}{12}}{2}

Apply the fraction rule:

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

= \frac{7 r}{12 \cdot 2}

Multiply the numbers:

12 \cdot 2 = 24

= \frac{7 r}{24}

= 2 cos (\frac{7 r}{24}) sin (\frac{24 x - r}{24})

= 2 cos (\frac{7 r}{24}) sin (\frac{24 x - r}{24})

2 cos (\frac{7 r}{24}) sin (\frac{24 x - r}{24}) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

sin (\frac{24 x - r}{24}) = 0

General solutions for

sin (\frac{24 x - r}{24}) = 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{24 x - r}{24} = 0 + 2 πn, \frac{24 x - r}{24} = π + 2 πn

\frac{24 x - r}{24} = 0 + 2 πn, \frac{24 x - r}{24} = π + 2 πn

Solve

\frac{24 x - r}{24} = 0 + 2 πn : x = 2 πn + \frac{r}{24}

\frac{24 x - r}{24} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{24 x - r}{24} = 2 πn

Multiply both sides by

24

\frac{24 x - r}{24} = 2 πn

Multiply both sides by

24

\frac{24 ( 24 x - r )}{24} = 24 \cdot 2 πn

Simplify

24 x - r = 48 πn

24 x - r = 48 πn

Move

r

to the right side

24 x - r = 48 πn

Add

r

to both sides

24 x - r + r = 48 πn + r

Simplify

24 x = 48 πn + r

24 x = 48 πn + r

Divide both sides by

24

24 x = 48 πn + r

Divide both sides by

24

\frac{24 x}{24} = \frac{48 πn}{24} + \frac{r}{24}

Simplify

x = 2 πn + \frac{r}{24}

x = 2 πn + \frac{r}{24}

Solve

\frac{24 x - r}{24} = π + 2 πn : x = π + 2 πn + \frac{r}{24}

\frac{24 x - r}{24} = π + 2 πn

Multiply both sides by

24

\frac{24 x - r}{24} = π + 2 πn

Multiply both sides by

24

\frac{24 ( 24 x - r )}{24} = 24 π + 24 \cdot 2 πn

Simplify

24 x - r = 24 π + 48 πn

24 x - r = 24 π + 48 πn

Move

r

to the right side

24 x - r = 24 π + 48 πn

Add

r

to both sides

24 x - r + r = 24 π + 48 πn + r

Simplify

24 x = 24 π + 48 πn + r

24 x = 24 π + 48 πn + r

Divide both sides by

24

24 x = 24 π + 48 πn + r

Divide both sides by

24

\frac{24 x}{24} = \frac{24 π}{24} + \frac{48 πn}{24} + \frac{r}{24}

Simplify

\frac{24 x}{24} = \frac{24 π}{24} + \frac{48 πn}{24} + \frac{r}{24}

Simplify

\frac{24 x}{24} : x

\frac{24 x}{24}

Divide the numbers:

\frac{24}{24} = 1

= x

Simplify

\frac{24 π}{24} + \frac{48 πn}{24} + \frac{r}{24} : π + 2 πn + \frac{r}{24}

\frac{24 π}{24} + \frac{48 πn}{24} + \frac{r}{24}

Divide the numbers:

\frac{24}{24} = 1

= π + \frac{48 πn}{24} + \frac{r}{24}

Divide the numbers:

\frac{48}{24} = 2

= π + 2 πn + \frac{r}{24}

x = π + 2 πn + \frac{r}{24}

x = π + 2 πn + \frac{r}{24}

x = π + 2 πn + \frac{r}{24}

x = 2 πn + \frac{r}{24}, x = π + 2 πn + \frac{r}{24}

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor x,sin(x+r/4)+sin(x-r/3)=0 ?
The general solution for solvefor x,sin(x+r/4)+sin(x-r/3)=0 is x=2pin+r/(24),x=pi+2pin+r/(24)