What is the general solution for solvefor y,sin(x+y)+sin(y+z)+sin(x+z)=0 ?

The general solution for solvefor y,sin(x+y)+sin(y+z)+sin(x+z)=0 is

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solvefor y,sin(x+y)+sin(y+z)+sin(x+z)=0

Solution

solve for

y, sin (x + y) + sin (y + z) + sin (x + z) = 0

Solution

y = arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}, y = π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

Solution steps

sin (x + y) + sin (y + z) + sin (x + z) = 0

Rewrite using trig identities

sin (x + y) + sin (y + z) + sin (x + z)

Use the Sum to Product identity:

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

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

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

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

Add similar elements:

y + y = 2 y

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

Expand

x + y - (y + z) : x - z

x + y - (y + z)

- (y + z) : - y - z

- (y + z)

Distribute parentheses

= - y - z

Apply minus-plus rules

+ (- a) = - a

= - y - z

= x + y - y - z

Add similar elements:

y - y = 0

= x - z

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

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

sin (x + z) + 2 cos (\frac{x - z}{2}) sin (\frac{x + 2 y + z}{2}) = 0

Move

sin (x + z)

to the right side

sin (x + z) + 2 cos (\frac{x - z}{2}) sin (\frac{x + 2 y + z}{2}) = 0

Subtract

sin (x + z)

from both sides

sin (x + z) + 2 cos (\frac{x - z}{2}) sin (\frac{x + 2 y + z}{2}) - sin (x + z) = 0 - sin (x + z)

Simplify

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

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

Divide both sides by

2 cos (\frac{x - z}{2}); x \neq = π + 4 πn + z, x \neq = 3 π + 4 πn + z

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

Divide both sides by

2 cos (\frac{x - z}{2}); x \neq = π + 4 πn + z, x \neq = 3 π + 4 πn + z

\frac{2 cos ( \frac{x - z}{2} ) sin ( \frac{x + 2 y + z}{2} )}{2 cos ( \frac{x - z}{2} )} = \frac{- sin ( x + z )}{2 cos ( \frac{x - z}{2} )}; x \neq = π + 4 πn + z, x \neq = 3 π + 4 πn + z

Simplify

sin (\frac{x + 2 y + z}{2}) = - \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}; x \neq = π + 4 πn + z, x \neq = 3 π + 4 πn + z

sin (\frac{x + 2 y + z}{2}) = - \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}; x \neq = π + 4 πn + z, x \neq = 3 π + 4 πn + z

Apply trig inverse properties

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

General solutions for

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

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

\frac{x + 2 y + z}{2} = arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn, \frac{x + 2 y + z}{2} = π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn

\frac{x + 2 y + z}{2} = arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn, \frac{x + 2 y + z}{2} = π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn

Solve

\frac{x + 2 y + z}{2} = arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn : y = arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

\frac{x + 2 y + z}{2} = arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn

Multiply both sides by

2

\frac{x + 2 y + z}{2} = arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn

Multiply both sides by

2

\frac{2 ( x + 2 y + z )}{2} = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 \cdot 2 πn

Simplify

\frac{2 ( x + 2 y + z )}{2} = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 \cdot 2 πn

Simplify

\frac{2 ( x + 2 y + z )}{2} : x + 2 y + z

\frac{2 ( x + 2 y + z )}{2}

Divide the numbers:

\frac{2}{2} = 1

= x + 2 y + z

Simplify

2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 \cdot 2 πn : 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn

2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

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

x + 2 y + z = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn

x + 2 y + z = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn

x + 2 y + z = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn

Move

x

to the right side

x + 2 y + z = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn

Subtract

x

from both sides

x + 2 y + z - x = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x

Simplify

2 y + z = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x

2 y + z = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x

Move

z

to the right side

2 y + z = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x

Subtract

z

from both sides

2 y + z - z = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x - z

Simplify

2 y = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x - z

2 y = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x - z

Divide both sides by

2

2 y = 2 arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x - z

Divide both sides by

2

\frac{2 y}{2} = \frac{2 arcsin ( - \frac{s i n ( x + z )}{2 c o s ( \frac{x - z}{2} )} )}{2} + \frac{4 πn}{2} - \frac{x}{2} - \frac{z}{2}

Simplify

\frac{2 y}{2} = \frac{2 arcsin ( - \frac{s i n ( x + z )}{2 c o s ( \frac{x - z}{2} )} )}{2} + \frac{4 πn}{2} - \frac{x}{2} - \frac{z}{2}

Simplify

\frac{2 y}{2} : y

\frac{2 y}{2}

Divide the numbers:

\frac{2}{2} = 1

= y

Simplify

\frac{2 arcsin ( - \frac{s i n ( x + z )}{2 c o s ( \frac{x - z}{2} )} )}{2} + \frac{4 πn}{2} - \frac{x}{2} - \frac{z}{2} : arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

\frac{2 arcsin ( - \frac{s i n ( x + z )}{2 c o s ( \frac{x - z}{2} )} )}{2} + \frac{4 πn}{2} - \frac{x}{2} - \frac{z}{2}

Divide the numbers:

\frac{2}{2} = 1

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

Divide the numbers:

\frac{4}{2} = 2

= arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

y = arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

y = arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

y = arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

Solve

\frac{x + 2 y + z}{2} = π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn : y = π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

\frac{x + 2 y + z}{2} = π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn

Multiply both sides by

2

\frac{x + 2 y + z}{2} = π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn

Multiply both sides by

2

\frac{2 ( x + 2 y + z )}{2} = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 \cdot 2 πn

Simplify

\frac{2 ( x + 2 y + z )}{2} = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 \cdot 2 πn

Simplify

\frac{2 ( x + 2 y + z )}{2} : x + 2 y + z

\frac{2 ( x + 2 y + z )}{2}

Divide the numbers:

\frac{2}{2} = 1

= x + 2 y + z

Simplify

2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 \cdot 2 πn : 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn

2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn

x + 2 y + z = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn

x + 2 y + z = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn

x + 2 y + z = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn

Move

x

to the right side

x + 2 y + z = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn

Subtract

x

from both sides

x + 2 y + z - x = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x

Simplify

2 y + z = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x

2 y + z = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x

Move

z

to the right side

2 y + z = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x

Subtract

z

from both sides

2 y + z - z = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x - z

Simplify

2 y = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x - z

2 y = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x - z

Divide both sides by

2

2 y = 2 π + 2 arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 4 πn - x - z

Divide both sides by

2

\frac{2 y}{2} = \frac{2 π}{2} + \frac{2 arcsin ( \frac{s i n ( x + z )}{2 c o s ( \frac{x - z}{2} )} )}{2} + \frac{4 πn}{2} - \frac{x}{2} - \frac{z}{2}

Simplify

\frac{2 y}{2} = \frac{2 π}{2} + \frac{2 arcsin ( \frac{s i n ( x + z )}{2 c o s ( \frac{x - z}{2} )} )}{2} + \frac{4 πn}{2} - \frac{x}{2} - \frac{z}{2}

Simplify

\frac{2 y}{2} : y

\frac{2 y}{2}

Divide the numbers:

\frac{2}{2} = 1

= y

Simplify

\frac{2 π}{2} + \frac{2 arcsin ( \frac{s i n ( x + z )}{2 c o s ( \frac{x - z}{2} )} )}{2} + \frac{4 πn}{2} - \frac{x}{2} - \frac{z}{2} : π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

\frac{2 π}{2} + \frac{2 arcsin ( \frac{s i n ( x + z )}{2 c o s ( \frac{x - z}{2} )} )}{2} + \frac{4 πn}{2} - \frac{x}{2} - \frac{z}{2}

Divide the numbers:

\frac{2}{2} = 1

= π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + \frac{4 πn}{2} - \frac{x}{2} - \frac{z}{2}

Divide the numbers:

\frac{4}{2} = 2

= π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

y = π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

y = π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

y = π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

y = arcsin (- \frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}, y = π + arcsin (\frac{sin ( x + z )}{2 cos ( \frac{x - z}{2} )}) + 2 πn - \frac{x}{2} - \frac{z}{2}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor y,sin(x+y)+sin(y+z)+sin(x+z)=0 ?
The general solution for solvefor y,sin(x+y)+sin(y+z)+sin(x+z)=0 is