Explicit solution of equations with trigonometric function [closed]

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How can an explicit solution for $x,y$ be obtained given
$$asin(x-y)=sin x$$
$$asin(y-x)=sin y$$

trigonometry systems-of-equations

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edited Jul 19 at 8:09

Parcly Taxel

33.6k136588

asked Jul 19 at 7:29

jarhead

1148

closed as off-topic by José Carlos Santos, Strants, Xander Henderson, Jose Arnaldo Bebita Dris, user223391 Jul 20 at 1:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Strants, Xander Henderson, Jose Arnaldo Bebita Dris, Community

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  As $arcsin(-a)=-arcsin a,$ we have $sin x=-sin y=sin(-y),x=npi+(-1)^n(-y)$
  – lab bhattacharjee
  Jul 19 at 7:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Is $asin(.)$ $sin^1(.)$ ?
  – Claude Leibovici
  Jul 19 at 7:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @labbhattacharjee, there is also a solution that depends on $a$
  – jarhead
  Jul 19 at 7:37
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

How can an explicit solution for $x,y$ be obtained given
$$asin(x-y)=sin x$$
$$asin(y-x)=sin y$$

trigonometry systems-of-equations

share|cite|improve this question

edited Jul 19 at 8:09

Parcly Taxel

33.6k136588

asked Jul 19 at 7:29

jarhead

1148

closed as off-topic by José Carlos Santos, Strants, Xander Henderson, Jose Arnaldo Bebita Dris, user223391 Jul 20 at 1:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Strants, Xander Henderson, Jose Arnaldo Bebita Dris, Community

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  As $arcsin(-a)=-arcsin a,$ we have $sin x=-sin y=sin(-y),x=npi+(-1)^n(-y)$
  – lab bhattacharjee
  Jul 19 at 7:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Is $asin(.)$ $sin^1(.)$ ?
  – Claude Leibovici
  Jul 19 at 7:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @labbhattacharjee, there is also a solution that depends on $a$
  – jarhead
  Jul 19 at 7:37
  

  
  
  
  

add a comment | 

up vote
1
down vote

favorite

up vote
1
down vote

favorite

How can an explicit solution for $x,y$ be obtained given
$$asin(x-y)=sin x$$
$$asin(y-x)=sin y$$

trigonometry systems-of-equations

share|cite|improve this question

edited Jul 19 at 8:09

Parcly Taxel

33.6k136588

asked Jul 19 at 7:29

jarhead

1148

How can an explicit solution for $x,y$ be obtained given
$$asin(x-y)=sin x$$
$$asin(y-x)=sin y$$

trigonometry systems-of-equations

share|cite|improve this question

edited Jul 19 at 8:09

Parcly Taxel

33.6k136588

asked Jul 19 at 7:29

jarhead

1148

share|cite|improve this question

share|cite|improve this question

edited Jul 19 at 8:09

Parcly Taxel

33.6k136588

edited Jul 19 at 8:09

Parcly Taxel

33.6k136588

edited Jul 19 at 8:09

Parcly Taxel

33.6k136588

33.6k136588

asked Jul 19 at 7:29

jarhead

1148

asked Jul 19 at 7:29

jarhead

1148

asked Jul 19 at 7:29

jarhead

1148

1148

closed as off-topic by José Carlos Santos, Strants, Xander Henderson, Jose Arnaldo Bebita Dris, user223391 Jul 20 at 1:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Strants, Xander Henderson, Jose Arnaldo Bebita Dris, Community

If this question can be reworded to fit the rules in the help center, please edit the question.

closed as off-topic by José Carlos Santos, Strants, Xander Henderson, Jose Arnaldo Bebita Dris, user223391 Jul 20 at 1:33

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – José Carlos Santos, Strants, Xander Henderson, Jose Arnaldo Bebita Dris, Community

If this question can be reworded to fit the rules in the help center, please edit the question.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  As $arcsin(-a)=-arcsin a,$ we have $sin x=-sin y=sin(-y),x=npi+(-1)^n(-y)$
  – lab bhattacharjee
  Jul 19 at 7:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Is $asin(.)$ $sin^1(.)$ ?
  – Claude Leibovici
  Jul 19 at 7:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @labbhattacharjee, there is also a solution that depends on $a$
  – jarhead
  Jul 19 at 7:37
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  As $arcsin(-a)=-arcsin a,$ we have $sin x=-sin y=sin(-y),x=npi+(-1)^n(-y)$
  – lab bhattacharjee
  Jul 19 at 7:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Is $asin(.)$ $sin^1(.)$ ?
  – Claude Leibovici
  Jul 19 at 7:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @labbhattacharjee, there is also a solution that depends on $a$
  – jarhead
  Jul 19 at 7:37
  

  
  
  
  

As $arcsin(-a)=-arcsin a,$ we have $sin x=-sin y=sin(-y),x=npi+(-1)^n(-y)$
– lab bhattacharjee
Jul 19 at 7:31

As $arcsin(-a)=-arcsin a,$ we have $sin x=-sin y=sin(-y),x=npi+(-1)^n(-y)$
– lab bhattacharjee
Jul 19 at 7:31

Is $asin(.)$ $sin^1(.)$ ?
– Claude Leibovici
Jul 19 at 7:31

Is $asin(.)$ $sin^1(.)$ ?
– Claude Leibovici
Jul 19 at 7:31

@labbhattacharjee, there is also a solution that depends on $a$
– jarhead
Jul 19 at 7:37

@labbhattacharjee, there is also a solution that depends on $a$
– jarhead
Jul 19 at 7:37

add a comment | 

1 Answer
1

active

oldest

votes

up vote
1
down vote



accepted

Obviously:

$$-sin x=sin y$$

It means that:

$$y=x + pi + 2kpi tag1$$

...or:

$$y=-x+2kpitag2$$

Case (1):

$$asin(x-(x+pi+2kpi))=-sin x$$

$$asin(-(2k+1)pi)=-sin x$$

$$0=sin x$$

$$x=npi, space y=npi+pi+2kpi=mpi$$

...for any $m,nin Z$

Case (2):

$$asin(x-(-x+2kpi)=sin x$$

$$asin(2x-2kpi)=sin x$$

$$asin(2x)=sin x$$

$$2asin x cos x=sin x$$

Trivial solution is $sin x=0$, but that solution is already obtained in case (1). Non-trivial solution is:

$$cos x=frac12a$$

$$x=pmarccosfrac12a + 2npi$$

$$y=-(pmarccosfrac12a + 2npi) + 2kpi$$

$$y=mparccosfrac12a + 2mpi$$

...for any $m,nin Z$. Obviously, this solution exists only for:

$$-1lefrac12ale1$$

share|cite|improve this answer

edited Jul 19 at 8:51

answered Jul 19 at 8:34

Oldboy

2,6101316

add a comment | 

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote



accepted

Obviously:

$$-sin x=sin y$$

It means that:

$$y=x + pi + 2kpi tag1$$

...or:

$$y=-x+2kpitag2$$

Case (1):

$$asin(x-(x+pi+2kpi))=-sin x$$

$$asin(-(2k+1)pi)=-sin x$$

$$0=sin x$$

$$x=npi, space y=npi+pi+2kpi=mpi$$

...for any $m,nin Z$

Case (2):

$$asin(x-(-x+2kpi)=sin x$$

$$asin(2x-2kpi)=sin x$$

$$asin(2x)=sin x$$

$$2asin x cos x=sin x$$

Trivial solution is $sin x=0$, but that solution is already obtained in case (1). Non-trivial solution is:

$$cos x=frac12a$$

$$x=pmarccosfrac12a + 2npi$$

$$y=-(pmarccosfrac12a + 2npi) + 2kpi$$

$$y=mparccosfrac12a + 2mpi$$

...for any $m,nin Z$. Obviously, this solution exists only for:

$$-1lefrac12ale1$$

share|cite|improve this answer

edited Jul 19 at 8:51

answered Jul 19 at 8:34

Oldboy

2,6101316

add a comment | 

up vote
1
down vote



accepted

Obviously:

$$-sin x=sin y$$

It means that:

$$y=x + pi + 2kpi tag1$$

...or:

$$y=-x+2kpitag2$$

Case (1):

$$asin(x-(x+pi+2kpi))=-sin x$$

$$asin(-(2k+1)pi)=-sin x$$

$$0=sin x$$

$$x=npi, space y=npi+pi+2kpi=mpi$$

...for any $m,nin Z$

Case (2):

$$asin(x-(-x+2kpi)=sin x$$

$$asin(2x-2kpi)=sin x$$

$$asin(2x)=sin x$$

$$2asin x cos x=sin x$$

Trivial solution is $sin x=0$, but that solution is already obtained in case (1). Non-trivial solution is:

$$cos x=frac12a$$

$$x=pmarccosfrac12a + 2npi$$

$$y=-(pmarccosfrac12a + 2npi) + 2kpi$$

$$y=mparccosfrac12a + 2mpi$$

...for any $m,nin Z$. Obviously, this solution exists only for:

$$-1lefrac12ale1$$

share|cite|improve this answer

edited Jul 19 at 8:51

answered Jul 19 at 8:34

Oldboy

2,6101316

add a comment | 

up vote
1
down vote



accepted

up vote
1
down vote



accepted

Obviously:

$$-sin x=sin y$$

It means that:

$$y=x + pi + 2kpi tag1$$

...or:

$$y=-x+2kpitag2$$

Case (1):

$$asin(x-(x+pi+2kpi))=-sin x$$

$$asin(-(2k+1)pi)=-sin x$$

$$0=sin x$$

$$x=npi, space y=npi+pi+2kpi=mpi$$

...for any $m,nin Z$

Case (2):

$$asin(x-(-x+2kpi)=sin x$$

$$asin(2x-2kpi)=sin x$$

$$asin(2x)=sin x$$

$$2asin x cos x=sin x$$

Trivial solution is $sin x=0$, but that solution is already obtained in case (1). Non-trivial solution is:

$$cos x=frac12a$$

$$x=pmarccosfrac12a + 2npi$$

$$y=-(pmarccosfrac12a + 2npi) + 2kpi$$

$$y=mparccosfrac12a + 2mpi$$

...for any $m,nin Z$. Obviously, this solution exists only for:

$$-1lefrac12ale1$$

share|cite|improve this answer

edited Jul 19 at 8:51

answered Jul 19 at 8:34

Oldboy

2,6101316

Obviously:

$$-sin x=sin y$$

It means that:

$$y=x + pi + 2kpi tag1$$

...or:

$$y=-x+2kpitag2$$

Case (1):

$$asin(x-(x+pi+2kpi))=-sin x$$

$$asin(-(2k+1)pi)=-sin x$$

$$0=sin x$$

$$x=npi, space y=npi+pi+2kpi=mpi$$

...for any $m,nin Z$

Case (2):

$$asin(x-(-x+2kpi)=sin x$$

$$asin(2x-2kpi)=sin x$$

$$asin(2x)=sin x$$

$$2asin x cos x=sin x$$

Trivial solution is $sin x=0$, but that solution is already obtained in case (1). Non-trivial solution is:

$$cos x=frac12a$$

$$x=pmarccosfrac12a + 2npi$$

$$y=-(pmarccosfrac12a + 2npi) + 2kpi$$

$$y=mparccosfrac12a + 2mpi$$

...for any $m,nin Z$. Obviously, this solution exists only for:

$$-1lefrac12ale1$$

share|cite|improve this answer

edited Jul 19 at 8:51

answered Jul 19 at 8:34

Oldboy

2,6101316

share|cite|improve this answer

share|cite|improve this answer

edited Jul 19 at 8:51

edited Jul 19 at 8:51

edited Jul 19 at 8:51

answered Jul 19 at 8:34

Oldboy

2,6101316

answered Jul 19 at 8:34

Oldboy

2,6101316

answered Jul 19 at 8:34

Oldboy

2,6101316

2,6101316

add a comment | 

add a comment |