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Math Help - Cosine identity

  1. #1
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    Cosine identity

    I came across the following while doing integration:
    Cosine identity-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
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  2. #2
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    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.
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  3. #3
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    Re: Cosine identity

    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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  4. #4
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    Re: Cosine identity

    cos(x+ y)= cos(x)cos(y)- sin(x)sin(y)

    So cos(x+\pi)= cos(x)cos(\pi)- sin(x)sin(\pi)= cos(x)(-1)+ sin(x)(0)= -cos(x)
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  5. #5
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    Re: Cosine identity

    Thanks to those who answered! An absurdly simple explanation.
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