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Math Help - addition theorems

  1. #1
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    addition theorems

    Hi!

    How do you prove that tan(∅+Φ) = (tan ∅ + tan Φ)/ (1- tanΦtan∅)?

    I'm able to prove the formulas for sin and cos (∅+Φ) but when i use sin/ cos = tan, it doesnt give me the above formula..

    thanks
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    Hi!

    How do you prove that tan(∅+Φ) = (tan ∅ + tan Φ)/ (1- tanΦtan∅)?

    I'm able to prove the formulas for sin and cos (∅+Φ) but when i use sin/ cos = tan, it doesnt give me the above formula..

    thanks
    This is really a topic for the trigonometry section. But all you need to know are the formulae for the sine and cosine of a sum.

    tan(A+B)=\frac{sin(A+B)}{cos(A+B)}=\frac{sinAcosB+  cosAsinB}{cosAcosB-sinAsinB}

    Divide the numerator and the denominator by cosAcosB

    \frac{\frac{sinA}{cosA}+\frac{sinB}{cosB}}{1-\frac{sinAsinB}{cosAcosB}}

    =\frac{tanA+tanB}{1-tan(A)tan(B)}
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  3. #3
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    Quote Originally Posted by alexandrabel90 View Post
    Hi!

    How do you prove that tan(∅+Φ) = (tan ∅ + tan Φ)/ (1- tanΦtan∅)?

    I'm able to prove the formulas for sin and cos (∅+Φ) but when i use sin/ cos = tan, it doesnt give me the above formula..

    thanks

    Of course it gives the above formula:

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

    Now simply multiply the right hand above by \displaystyle{\frac{\frac{1}{\cos x\cos y}}{\frac{1}{\cos x\cos y}}=1} and develop.

    Tonio
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