This week’s math fact of the week is an interesting trigonometric identity involving the tangent function that I happened across the other day while reading the Polymathematics archives. Given a triangle with angles A, B, and C:
That’s a pretty neat identity! But how can we show it to be true? You can read a simple proof after the jump.
The first thing we need to note is that the sum of the angles of a triangle is always \pi radians:
The other piece of information we’ll need to prove this identity is the following trigonometric identity:
Because A + B + C sum to \pi, we must have that tan C = tan (\pi – (A + B)). Applying the identity above to this yields the following:
Of course sin \pi = tan \pi = 0, so we can simplify this to:
Now let’s apply the identity again, this time to the result we just obtained:
Some elementary algebra — many thanks to Eric Lahtinen for pointing out that working backwards was the best way to proceed here — yields the following:
![$\tan A \cdot \tan B = 1 - \frac{\tan A + ]tan B}{\tan (A + B)}$](http://www.bostongeek.com/wp-content/wmf/wmf20060913-7.png)
If we multiply both sides by tan C, then we obtain:
Distributing the term on the right yields:
But as we showed earlier, tan C = – tan (A + B). QED.
Well that wraps up another weekly math fact. See you next time, and until then, don’t forget to send your math questions to jason AT bostongeek DOT com!
Aaaaaand…. lost me.
I don’t even read this because I know if will do nothing but give me a headache and anger me.
Carl, where did we lose you? I can explain in greater detail if you’d like.
You lost me at… ‘…interesting trignometric identity…’
Alas, my interest (and apparent capacity) for higher mathematical functions waned in about the 10th grade. Gimme sciences like Biology, Chemistry, even a little Physics (right up until the heavy math starts), and I’m okay. Give me History and Literature…
But Trig? I’d rather be repeatedly beaten with the book than read it.
I remember tangents… vaguely. Like awakening from a dream, struggling to remember details that slip through my fingers like so many grains of sand. Bitter, bitter sand.
From the look of the pictures, isn’t that what happened to you last wekend? The bitter sand was probably just from something on the sidewalk.
Only thing that slowed me down was not-reading -= as =-, J, if that helps.
(also, sorry I never got chance to look this over for you)
Look, we all know the only way I am going to understand this is if someone can explain it using pr0n or perhaps D&D.
So… say I have a pile of magazines and an Orc double-axe… work with me here…
And bite me, Remy.
Not my job, Sidewalk Sleeper.
Dave wrote,
No worries! As I mention in the post, Eric Lahtinen took one look at my first attempt and laughed, then showed me the easy way. I had been trying to work from tan C = -tan (A + B) by substituting into tan A + tan B + tan C (so tan A + tan B – tan (A + B)), then substituting into that equation using the identity (tan A + tan B – {tan A + tan B}/{1 – (tan A * tan B)}), etc.
I ended up with a whole page of algebra and not much to show for it…
Very super information.