Expert: Paul Klarreich - 4/16/2007

Question
Can you, perhaps, help me anyway ?

Da:  <donotreply@allexperts.com>
Inviato:  domenica 15 aprile 2007 12.40.45
A:  dumbed-down@hotmail.it
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We are sorry, but Andrell wasn't able to answer your question:

Simplify the following :

sin (90°+ x)sin( 180° + x) + cos (90° + x ) cos( 180° - x )

Any help, please ?

Thank you

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Thank you !  

Answer
Questioner:   Mark
Category:  Advanced Math
 
Subject:  Plane Trigonometry
Question:  Can you, perhaps, help me anyway ?

Da:  <donotreply@allexperts.com>
Inviato:  domenica 15 aprile 2007 12.40.45
A:  dumbed-down@hotmail.it
Oggetto:  AllExperts Reply to your question

|  |  | Posta in arrivo


We are sorry, but Andrell wasn't able to answer your question:

Simplify the following :

sin (90°+ x)sin( 180° + x) + cos (90° + x ) cos( 180° - x )

Any help, please ?

Thank you

Unfortunately, not all volunteers are able to answer all questions. Feel free to
repost your question to another volunteer by clicking on the link
http://www.allexperts.com/browse.cgi?catLvl=3&catID=657

allexperts.com


Thank you !
..................................
Hi, Mark,

I don't understand all the email stuff, but:

sin (90°+ x)sin( 180° + x) + cos (90° + x ) cos( 180° - x )

should be simplifiable by expanding each of the functions.
(see the note at the end about 'Maybe you mistyped.')

[I shall omit the degree symbols in the typing.]

We will use these facts:
sin 90 = 1
cos 90 = 0
sin 180 = 0
cos 180 = -1


sin(90 + x) = sin 90 cos x + cos 90 sin x = cos x
sin(180 + x) = sin 180 cos x + cos 180 sin x = - sin x
cos(90 + x) = cos 90 cos x - sin 90 sin x = - sin x
cos(180 - x) = cos 180 cos x + sin 180 sin x = - cos x

Your expression is now:

cos x (-sin x) + (- sin x)(- cos x)
= - cos x sin x + sin x cos x
= 0
..............................
BTW, did you mistype it?  Should it have been:

sin (90°+ x)sin( 180° + x) + cos (90° + x ) cos( 180° + x )
[cos(180+x) instead of cos(180-x)]

In that case, you could recognize it as an example of

cos(A-B) = cos A cos B + sin A sin B
and write cos(180 + x - (90 + x)) = cos(180 + x - 90 - x) = cos 90 = 0.