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Math Help - tan, cot in terms of p,q

  1. #1
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    tan, cot in terms of p,q

    Given that sin a/sin b = p and cos a/cos b = q, express tan a and cot a in terms of p and q. (a and b are acute angles). I have no idea where to start on this one. I've tried a few things but always get stuck.
    Thanks much.
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  2. #2
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    Quote Originally Posted by daydreembelievr View Post
    Given that sin a/sin b = p and cos a/cos b = q, express tan a and cot a in terms of p and q. (a and b are acute angles). I have no idea where to start on this one. I've tried a few things but always get stuck.
    Thanks much.
    Here's a big start:


    \frac{\sin a}{\sin b} = p \Rightarrow \sin b = \frac{\sin a}{p} .... (1)


    \frac{\cos a}{\cos b} = q \Rightarrow \cos b = \frac{\cos a}{q} .... (2)

    Square both equations and add:

    1 = \frac{\sin^2 a}{p^2} + \frac{\cos^2 a}{q^2} \Rightarrow p^2 q^2 = q^2 \sin^2 a + p^2 \cos^2 a

    \Rightarrow \frac{p^2 q^2 }{\cos^2 a} = q^2 \tan^2 a + p^2 .... (3)


    Substitute the identity 1 + \tan^2 a = \frac{1}{\cos^2 a} into equation (3) and solve for \tan^2 a and hence \tan a.
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