# Cosine identity

• Apr 7th 2013, 01:17 PM
Giestforlife
Cosine identity
I came across the following while doing integration:
Attachment 27849

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
• Apr 7th 2013, 01:38 PM
zhandele
Re: Cosine identity
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.
• Apr 7th 2013, 01:41 PM
Shakarri
Re: Cosine identity
If you draw the unit circle and label the quadrants with which trig values are positive it is easy to visualise.
http://puu.sh/2vSqY

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)
• Apr 7th 2013, 01:53 PM
HallsofIvy
Re: Cosine identity
cos(x+ y)= cos(x)cos(y)- sin(x)sin(y)

So $\displaystyle cos(x+\pi)= cos(x)cos(\pi)- sin(x)sin(\pi)= cos(x)(-1)+ sin(x)(0)= -cos(x)$
• Apr 10th 2013, 12:34 PM
Giestforlife
Re: Cosine identity
Thanks to those who answered! An absurdly simple explanation.