SOLUTION: How do I express sin(x+y+z) in terms of cosines and sines of x, y and z? I know that you can express sin(x+y) into =sinxcosy+cosxsiny

Algebra ->  Expressions-with-variables -> SOLUTION: How do I express sin(x+y+z) in terms of cosines and sines of x, y and z? I know that you can express sin(x+y) into =sinxcosy+cosxsiny       Log On


   

Question 920256: How do I express sin(x+y+z) in terms of cosines and sines of x, y and z?
I know that you can express
sin(x+y) into
=sinxcosy+cosxsiny
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Let Q+=+x%2By. You don't have to use Q. Just any other variable not being used.

sin%28x%2By%2Bz%29 will then turn into sin%28Q%2Bz%29 after you replace "x+y" with "Q" (the substitution is valid based on the definition we made above).

Expand out sin%28Q%2Bz%29:

sin%28Q%2Bz%29+=+sin%28Q%29cos%28z%29%2Bcos%28Q%29sin%28z%29

Now we have sin%28Q%29cos%28z%29%2Bcos%28Q%29sin%28z%29. Next, replace every Q with "x+y" to get

sin%28Q%29cos%28z%29%2Bcos%28Q%29sin%28z%29

sin%28x%2By%29cos%28z%29%2Bcos%28x%2By%29sin%28z%29 replace every Q with "x+y"

The next step is to expand sin%28x%2By%29 and cos%28x%2By%29

sin%28x%2By%29+=+sin%28x%29cos%28y%29%2Bcos%28x%29sin%28y%29
cos%28x%2By%29+=+cos%28x%29cos%28y%29-sin%28x%29sin%28y%29

which you will use to replace sin%28x%2By%29 and cos%28x%2By%29 in sin%28x%2By%29cos%28z%29%2Bcos%28x%2By%29sin%28z%29

sin%28x%2By%29cos%28z%29%2Bcos%28x%2By%29sin%28z%29

%28sin%28x%29cos%28y%29%2Bcos%28x%29sin%28y%29%29cos%28z%29%2Bcos%28x%2By%29sin%28z%29 Replace sin%28x%2By%29 with sin%28x%29cos%28y%29%2Bcos%28x%29sin%28y%29

Replace cos%28x%2By%29 with cos%28x%29cos%28y%29-sin%28x%29sin%28y%29

Distribute


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So sin%28x%2By%2Bz%29 expands out into


Let me know if you need more help or if you need me to explain a step in more detail.
Feel free to email me at jim_thompson5910@hotmail.com
or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html

Thanks,

Jim