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Math Help - Tangents

  1. #1
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    Tangents

    If A+B+C=90^\circ, show that tanAtanB+tanBtanC+tanCtanA=1
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    Hello Punch
    Quote Originally Posted by Punch View Post
    If A+B+C=90^\circ, show that tanAtanB+tanBtanC+tanCtanA=1
    A+B+C =90^o

    \Rightarrow A+B = 90^o-C

    \Rightarrow \tan(A+B) = \tan(90^o-C)

    \Rightarrow \frac{\tan A + \tan B}{1-\tan A\tan B}= \cot C

    I think you can finish up now.

    Grandad
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  3. #3
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    Thanks but I don't understand how to relate them..
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    Hello Punch

    Here's the last bit, then:

    \frac{\tan A + \tan B}{1-\tan A\tan B}= \cot C
    =\frac{1}{\tan C}
    \Rightarrow \tan C(\tan A + \tan B) = 1 - \tan A \tan B

    \Rightarrow \tan A\tan B + \tan B\tan C +\tan C \tan A = 1

    Grandad
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    It is all so confusing... I have tried to combine the 2 posts but it makes no sense to me.

    If A+B+C=90^\circ, show that tanAtanB+tanBtanC+tanCtanA=1

    How about telling me the idea of solving this question? Is it about changing everything in terms of A?
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  6. #6
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    Hello Punch
    Quote Originally Posted by Punch View Post
    It is all so confusing... I have tried to combine the 2 posts but it makes no sense to me.

    If A+B+C=90^\circ, show that tanAtanB+tanBtanC+tanCtanA=1

    How about telling me the idea of solving this question? Is it about changing everything in terms of A?
    I'm not sure what you mean by this. You were asked to show that if a certain statement was true (namely, A+B+C=90^o) then another statement follows (namely, \tan A\tan B+\tan B\tan C+\tan C\tan A=1).

    I have done this by starting with the equation A+B+C=90^o and ending with the equation \tan A\tan B+\tan B\tan C+\tan C\tan A=1.

    Putting together my two posts, we get:
    A+B+C=90^o
    \Rightarrow A+B = 90^o-C

    \Rightarrow \tan(A+B) = \tan(90^o-C)

    \Rightarrow \frac{\tan A + \tan B}{1-\tan A\tan B}= \cot C
    =\frac{1}{\tan C}
    \Rightarrow \tan C(\tan A + \tan B) = 1 - \tan A \tan B

    \Rightarrow \tan A\tan B + \tan B\tan C +\tan C \tan A = 1
    So we have now done everything we were asked to do. Are there any steps in the working that you don't understand?

    Grandad
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  7. #7
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    I don't understand <br />
\Rightarrow \tan A\tan B + \tan B\tan C +\tan C \tan A = 1<br />
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  8. #8
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    Expand the brackets and then add \tan{A}\tan{B} to both sides...
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    To solve this question, I have to relate it to the tangent additional formulae.
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  10. #10
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    Hello Punch
    Quote Originally Posted by Punch View Post
    To solve this question, I have to relate it to the tangent additional formulae.
    Yes, and you'll see that I have used the tangent addition formula where \tan(A+B) is written on the following line as \frac{\tan A + \tan B}{1-\tan A \tan B}.

    Grandad
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  11. #11
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    Quote Originally Posted by Grandad View Post
    Hello PunchYes, and you'll see that I have used the tangent addition formula where \tan(A+B) is written on the following line as \frac{\tan A + \tan B}{1-\tan A \tan B}.

    Grandad
    I am sorry to be posting on an "old" thread. However, I was wondering how I could know when to relate such questions to formulas.

    Thanks in advance.(Enlarged for Grandad)
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  12. #12
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    Hello Punch
    Quote Originally Posted by Punch View Post
    I am sorry to be posting on an "old" thread. However, I was wondering how I could know when to relate such questions to formulas.

    Thanks in advance.(Enlarged for Grandad)
    Thanks for the concession to my eyesight!

    As far as your question is concerned, I'm afraid that experience - practice, practice, practice - is the only answer: just like learning to play a musical instrument.

    There are just so many trig formulae, it's impossible to give any simple answer as to what to use and when. Since there are so many formulae, I always found it helpful myself - and so I taught my students - to recite the formulae aloud that I was trying to learn. Write them on Post-It stickers, and stick them up in places where you'll see them while you're doing something else - something that doesn't require much brain power (peeling potatoes, painting a fence, listening to your girlfriend, etc, etc) and recite them over and over. (Perhaps I wasn't really serious about the girlfriend bit...)

    Grandad
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