Cosine identity

Jul 2012
25
0
NOVA
I came across the following while doing integration:
CodeCogsEqn.gif

I don't need a proof, I just want to understand what general property (law, theorem, identity?) of trig that is at work here.

Thanks,

Giest
 
Nov 2012
197
30
Normal, IL USA
Think of the unit circle. You'll see that cos(x + pi) = - cos(x). So cos(x) - cos(x + pi) = cos(x) - (-cos(x)) = 2 cos(x). Then divide that by 2.
 
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Oct 2012
751
212
Ireland
If you draw the unit circle and label the quadrants with which trig values are positive it is easy to visualise.


You can see that adding 180 degrees onto an angle brings a point on the circle from a positive quadrant to a negative quadrant or vice versa. But when you add 180 degrees to a point it remains the same angle from the horizontal axis.
Taking these two factors into account you can see that cos(x)= -cos(x+180)
 
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HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
cos(x+ y)= cos(x)cos(y)- sin(x)sin(y)

So \(\displaystyle cos(x+\pi)= cos(x)cos(\pi)- sin(x)sin(\pi)= cos(x)(-1)+ sin(x)(0)= -cos(x)\)
 
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Jul 2012
25
0
NOVA
Thanks to those who answered! An absurdly simple explanation.