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Simplify trigonometric expression using trigonometric identities

Simplify a trigonometric expressionbest way to detect the trigonometric identites that shall work on a given expression so as to simplify it accordingly?Trigonometric Identities HWsimplyfing trigonometric expression using phase shift identitiesSimplify the expression using trigonometric identitiesHow to simplify trigonometric expressionSimplify Trigonometric ExpressionHow to simplify a trigonometric expression with the identitiesTrigonometric IdentitiesHow to evaluate this trigonometric expression

3

$begingroup$

I have the trigonometric expression: $$2sin x +2sin left(fracpi 3 -xright) $$
and it should simplified in: $$sin x + sqrt 3 cos x$$ but I do not know what formulas to apply. Could you tell me how to simplify it?

trigonometry

share|cite|improve this question

edited Apr 1 at 15:55

MarianD

2,1711618

asked Apr 1 at 15:19

zaz9999zaz9999

162

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zaz9999 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  mathworld.wolfram.com/ProsthaphaeresisFormulas.html
  $endgroup$
  – lab bhattacharjee
  Apr 1 at 15:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Just between us: IMO the two expressions are of the same complexity.
  $endgroup$
  – Yves Daoust
  Apr 1 at 15:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  = 2 cos(x-π/6).
  $endgroup$
  – Anton Sherwood
  Apr 1 at 22:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
  $endgroup$
  – dantopa
  Apr 2 at 0:58
  

  
  
  
  

add a comment | 

3

$begingroup$

I have the trigonometric expression: $$2sin x +2sin left(fracpi 3 -xright) $$
and it should simplified in: $$sin x + sqrt 3 cos x$$ but I do not know what formulas to apply. Could you tell me how to simplify it?

trigonometry

share|cite|improve this question

edited Apr 1 at 15:55

MarianD

2,1711618

asked Apr 1 at 15:19

zaz9999zaz9999

162

New contributor

zaz9999 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  mathworld.wolfram.com/ProsthaphaeresisFormulas.html
  $endgroup$
  – lab bhattacharjee
  Apr 1 at 15:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Just between us: IMO the two expressions are of the same complexity.
  $endgroup$
  – Yves Daoust
  Apr 1 at 15:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  = 2 cos(x-π/6).
  $endgroup$
  – Anton Sherwood
  Apr 1 at 22:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer.
  $endgroup$
  – dantopa
  Apr 2 at 0:58
  

  
  
  
  

add a comment | 

$begingroup$

I have the trigonometric expression: $$2sin x +2sin left(fracpi 3 -xright) $$
and it should simplified in: $$sin x + sqrt 3 cos x$$ but I do not know what formulas to apply. Could you tell me how to simplify it?

trigonometry

share|cite|improve this question

edited Apr 1 at 15:55

MarianD

2,1711618

asked Apr 1 at 15:19

zaz9999zaz9999

162

New contributor

zaz9999 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited Apr 1 at 15:55

MarianD

2,1711618

asked Apr 1 at 15:19

zaz9999zaz9999

162

New contributor

zaz9999 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited Apr 1 at 15:55

MarianD

2,1711618

asked Apr 1 at 15:19

zaz9999zaz9999

162

New contributor

zaz9999 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

edited Apr 1 at 15:55

MarianD

2,1711618

edited Apr 1 at 15:55

MarianD

2,1711618

asked Apr 1 at 15:19

zaz9999zaz9999

162

New contributor

zaz9999 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked Apr 1 at 15:19

zaz9999zaz9999

162

3 Answers
3

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7

$begingroup$

Using the formula for $sin (alpha - beta)$ you obtain

beginalign
2&sin x +2sin left(fracpi 3 -xright)\
= 2&sin x +2left[sin left(fracpi 3right) cos x - cosleft(fracpi 3right) sin xright]\
= 2&sin x + 2left[sqrt 3 over 2 cos x - frac 1 2 sin xright]\[1ex]
= 2&sin x + sqrt 3 cos x - sin x\[1em] = , &colorredsin x + sqrt 3 cos x
endalign

share|cite|improve this answer

edited Apr 1 at 15:59

answered Apr 1 at 15:44

MarianDMarianD

2,1711618

$endgroup$

add a comment | 

4

$begingroup$

Use that $$sin(x-y)=sin(x)cos(y)-sin(y)cos(x)$$

share|cite|improve this answer

answered Apr 1 at 15:26

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

78.4k42867

$endgroup$

add a comment | 

1

$begingroup$

Hint: $sin(fracpi3-x)=sin fracpi3cos x-cos fracpi3sin x$

share|cite|improve this answer

answered Apr 1 at 15:25

VasyaVasya

4,2771618

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add a comment | 

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7

$begingroup$

Using the formula for $sin (alpha - beta)$ you obtain

beginalign
2&sin x +2sin left(fracpi 3 -xright)\
= 2&sin x +2left[sin left(fracpi 3right) cos x - cosleft(fracpi 3right) sin xright]\
= 2&sin x + 2left[sqrt 3 over 2 cos x - frac 1 2 sin xright]\[1ex]
= 2&sin x + sqrt 3 cos x - sin x\[1em] = , &colorredsin x + sqrt 3 cos x
endalign

share|cite|improve this answer

edited Apr 1 at 15:59

answered Apr 1 at 15:44

MarianDMarianD

2,1711618

$endgroup$

add a comment | 

7

$begingroup$

Using the formula for $sin (alpha - beta)$ you obtain

beginalign
2&sin x +2sin left(fracpi 3 -xright)\
= 2&sin x +2left[sin left(fracpi 3right) cos x - cosleft(fracpi 3right) sin xright]\
= 2&sin x + 2left[sqrt 3 over 2 cos x - frac 1 2 sin xright]\[1ex]
= 2&sin x + sqrt 3 cos x - sin x\[1em] = , &colorredsin x + sqrt 3 cos x
endalign

share|cite|improve this answer

edited Apr 1 at 15:59

answered Apr 1 at 15:44

MarianDMarianD

2,1711618

$endgroup$

add a comment | 

$begingroup$

Using the formula for $sin (alpha - beta)$ you obtain

beginalign
2&sin x +2sin left(fracpi 3 -xright)\
= 2&sin x +2left[sin left(fracpi 3right) cos x - cosleft(fracpi 3right) sin xright]\
= 2&sin x + 2left[sqrt 3 over 2 cos x - frac 1 2 sin xright]\[1ex]
= 2&sin x + sqrt 3 cos x - sin x\[1em] = , &colorredsin x + sqrt 3 cos x
endalign

share|cite|improve this answer

edited Apr 1 at 15:59

answered Apr 1 at 15:44

MarianDMarianD

2,1711618

$endgroup$

share|cite|improve this answer

edited Apr 1 at 15:59

answered Apr 1 at 15:44

MarianDMarianD

2,1711618

answered Apr 1 at 15:44

MarianDMarianD

2,1711618

answered Apr 1 at 15:44

MarianDMarianD

2,1711618

4

$begingroup$

Use that $$sin(x-y)=sin(x)cos(y)-sin(y)cos(x)$$

share|cite|improve this answer

answered Apr 1 at 15:26

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

78.4k42867

$endgroup$

add a comment | 

4

$begingroup$

Use that $$sin(x-y)=sin(x)cos(y)-sin(y)cos(x)$$

share|cite|improve this answer

answered Apr 1 at 15:26

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

78.4k42867

$endgroup$

add a comment | 

$begingroup$

Use that $$sin(x-y)=sin(x)cos(y)-sin(y)cos(x)$$

share|cite|improve this answer

answered Apr 1 at 15:26

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

78.4k42867

$endgroup$

share|cite|improve this answer

answered Apr 1 at 15:26

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

78.4k42867

answered Apr 1 at 15:26

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

78.4k42867

answered Apr 1 at 15:26

Dr. Sonnhard GraubnerDr. Sonnhard Graubner

78.4k42867

1

$begingroup$

Hint: $sin(fracpi3-x)=sin fracpi3cos x-cos fracpi3sin x$

share|cite|improve this answer

answered Apr 1 at 15:25

VasyaVasya

4,2771618

$endgroup$

add a comment | 

1

$begingroup$

Hint: $sin(fracpi3-x)=sin fracpi3cos x-cos fracpi3sin x$

share|cite|improve this answer

answered Apr 1 at 15:25

VasyaVasya

4,2771618

$endgroup$

add a comment | 

$begingroup$

Hint: $sin(fracpi3-x)=sin fracpi3cos x-cos fracpi3sin x$

share|cite|improve this answer

answered Apr 1 at 15:25

VasyaVasya

4,2771618

$endgroup$

share|cite|improve this answer

answered Apr 1 at 15:25

VasyaVasya

4,2771618

answered Apr 1 at 15:25

VasyaVasya

4,2771618

answered Apr 1 at 15:25

VasyaVasya

4,2771618