Tangents

**Punch** · January 28th 2010, 01:38 AM

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

**Grandad** · January 28th 2010, 04:47 AM

Hello Punch

Originally Posted by Punch

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

**Punch** · January 28th 2010, 11:58 PM

Thanks but I don't understand how to relate them..

**Grandad** · January 29th 2010, 12:36 AM

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

**Punch** · January 30th 2010, 06:12 AM

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?

**Grandad** · January 30th 2010, 10:22 AM

Hello Punch

Originally Posted by Punch

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

**Punch** · January 30th 2010, 07:18 PM

I don't understand $<br /> \Rightarrow \tan A\tan B + \tan B\tan C +\tan C \tan A = 1<br />$

**Prove It** · January 30th 2010, 07:24 PM

Expand the brackets and then add $\tan{A}\tan{B}$ to both sides...

**Punch** · February 1st 2010, 05:03 AM

To solve this question, I have to relate it to the tangent additional formulae.

**Grandad** · February 1st 2010, 06:03 AM

Hello Punch

Originally Posted by Punch

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

**Punch** · February 3rd 2010, 04:05 AM

Originally Posted by Grandad

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)

**Grandad** · February 3rd 2010, 04:24 AM

Hello Punch

Originally Posted by Punch

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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