SOLUTION: How do I simplify this? Working with radians here. {{{sin(pi/2-x) +sin(pi-x)+sin(3pi/2-x)+sin(2pi-x)}}}

Algebra ->  Expressions-with-variables -> SOLUTION: How do I simplify this? Working with radians here. {{{sin(pi/2-x) +sin(pi-x)+sin(3pi/2-x)+sin(2pi-x)}}}      Log On


   

Question 920239: How do I simplify this?
Working with radians here.
sin%28pi%2F2-x%29+%2Bsin%28pi-x%29%2Bsin%283pi%2F2-x%29%2Bsin%282pi-x%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To start things off, I'm going to expand out each piece being added


Expand sin%28pi%2F2-x%29:
sin%28pi%2F2-x%29+=+sin%28pi%2F2%29cos%28x%29-cos%28pi%2F2%29sin%28x%29
sin%28pi%2F2-x%29+=+1%2Acos%28x%29-0%2Asin%28x%29
sin%28pi%2F2-x%29+=+cos%28x%29



Expand sin%28pi-x%29
sin%28pi-x%29=sin%28pi%29cos%28x%29-cos%28pi%29sin%28x%29
sin%28pi-x%29=0%2Acos%28x%29-%28-1%29sin%28x%29
sin%28pi-x%29=sin%28x%29



Expand sin%283pi%2F2-x%29
sin%283pi%2F2-x%29+=+sin%283pi%2F2%29cos%28x%29-cos%283pi%2F2%29sin%28x%29
sin%283pi%2F2-x%29+=+-1%2Acos%28x%29-0%2Asin%28x%29
sin%283pi%2F2-x%29+=+-cos%28x%29



Expand sin%282pi-x%29
sin%282pi-x%29=sin%282pi%29cos%28x%29-cos%282pi%29sin%28x%29
sin%282pi-x%29=0%2Acos%28x%29-1%2Asin%28x%29
sin%282pi-x%29=-sin%28x%29


-------------------------------------------------------

So we know the following:


sin%28pi%2F2-x%29+=+cos%28x%29
sin%28pi-x%29=sin%28x%29
sin%283pi%2F2-x%29+=+-cos%28x%29
sin%282pi-x%29=-sin%28x%29

-------------------------------------------------------

Which will be used below

sin%28pi%2F2-x%29+%2Bsin%28pi-x%29%2Bsin%283pi%2F2-x%29%2Bsin%282pi-x%29

cos%28x%29+%2Bsin%28pi-x%29%2Bsin%283pi%2F2-x%29%2Bsin%282pi-x%29 Replace sin%28pi%2F2-x%29 with cos%28x%29 (since sin%28pi%2F2-x%29+=+cos%28x%29)

cos%28x%29+%2Bsin%28x%29%2Bsin%283pi%2F2-x%29%2Bsin%282pi-x%29 Replace sin%28pi-x%29 with sin%28x%29 (since sin%28pi-x%29+=+sin%28x%29)

cos%28x%29+%2Bsin%28x%29%2B%28-cos%28x%29%29%2Bsin%282pi-x%29 Replace sin%283pi%2F2-x%29 with -cos%28x%29 (since sin%283pi%2F2-x%29+=+-cos%28x%29)

cos%28x%29+%2Bsin%28x%29%2B%28-cos%28x%29%29%2B%28-sin%28x%29%29 Replace sin%282pi-x%29 with -sin%28x%29 (since sin%282pi-x%29+=+-sin%28x%29)

cos%28x%29+%2Bsin%28x%29-cos%28x%29-sin%28x%29

%28cos%28x%29-cos%28x%29%29%2B%28sin%28x%29-sin%28x%29%29

%280%29%2B%280%29

0%2B0

0

So in the end, sin%28pi%2F2-x%29+%2Bsin%28pi-x%29%2Bsin%283pi%2F2-x%29%2Bsin%282pi-x%29 simplifies to 0

In other words, sin%28pi%2F2-x%29+%2Bsin%28pi-x%29%2Bsin%283pi%2F2-x%29%2Bsin%282pi-x%29=0 is an identity (true for all real numbers x)


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