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Math Help - Figuring out trig identities

  1. #1
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    Figuring out trig identities

    So, I know that from the sine, cosine, and tangent of a sum or difference I can figure out the double angle and half angle identities for a sine, cosine, and tangent.

    I'm having trouble finding a pattern to memorize the tangent of a sum or difference, so I've done nothing with the tangents yet.

    But I'm successfully able to figure out the double angle identities using the sum identities.

    I haven't figured out what to do to get to the half angle identities from the sum or difference identities or the double angle identities.

    So, is there any way to figure out the tangent of a sum or difference from the sine and cosine of a sum or difference, and is there a way to get to the half angle identities from the sum/difference identities?

    Thanks!
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by Wolvenmoon View Post
    So, I know that from the sine, cosine, and tangent of a sum or difference I can figure out the double angle and half angle identities for a sine, cosine, and tangent.

    I'm having trouble finding a pattern to memorize the tangent of a sum or difference, so I've done nothing with the tangents yet.

    But I'm successfully able to figure out the double angle identities using the sum identities.

    I haven't figured out what to do to get to the half angle identities from the sum or difference identities or the double angle identities.

    So, is there any way to figure out the tangent of a sum or difference from the sine and cosine of a sum or difference, and is there a way to get to the half angle identities from the sum/difference identities?

    Thanks!
    \tan (x+y) = \frac{\tan x+ \tan y }{1-\tan x \tan y }

    try to prove it using

    \tan (x+y) = \frac{\sin (x+y)}{\cos (x+y) }

    and you said you know the identity of the sum for sin and cos

    I believe that if you want to memorize a theorem or a identity try to prove it in your way

    for the half identity for example

    \sin 2x = 2\sin x \cos x

    \sin x  = \frac{\sin 2x }{2\cos x }
    but

    \cos 2x = 2\cos^2 x -1 \Rightarrow \cos ^2 x = \frac{\cos 2x+1 }{2} \Rightarrow \cos x = \sqrt{\frac{\cos 2x +1 }{2}}

    \sin x = \frac{\sin 2x \cdot \sqrt{2}}{\sqrt{\cos 2x +1 }}

    I think it is useless to memorize it

    I'm sure that it is enough to memorize this ( in my opinion)

    \tan x = \frac{\sin x }{\cos x}

    \sin (x\mp y) = ??
    \cos(x\mp y)=??
    \sin 2x =??
    \cos 2x =?? or ?? or ??
    \sin ^2 x + \cos ^2 x
    from these you can find all identities
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  3. #3
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    After reading your post, I think I can get the tangent identities pretty easily now. The only place I'm having trouble with now is getting from trigfunc(2x) or trigfunc(x+y) to trigfunc(x/2) where trigfunc can be sin or cos.

    Thanks!
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