SOLUTION: Given three angles A, B, and C whose sum is 180 degrees. If tan A + tan B + tan C = x. Find the value of the product of the three tangents.

Algebra ->  Trigonometry-basics -> SOLUTION: Given three angles A, B, and C whose sum is 180 degrees. If tan A + tan B + tan C = x. Find the value of the product of the three tangents.      Log On


   

Question 1161717: Given three angles A, B, and C whose sum is 180 degrees. If tan A + tan B + tan C = x. Find the value of the product of the three tangents.
Found 2 solutions by KMST, Edwin McCravy:
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
We are told that A%2BB%2BC=180%5Eo , so A%2BB=180%5Eo-C
We (should) know that tan%28180%5Eo-C%29=-tan%28C%29 , so tan%28A%2BB%29=-tan%28C%29 .
According to the list of trigonometric identities that I found online
tan%28A%2BB%29=%28tan%28A%29%2Btan%28B%29%29%2F%281-tan%28A%29tan%28B%29%29 , so
-tan%28C%29=%28tan%28A%29%2Btan%28B%29%29%2F%281-tan%28A%29tan%28B%29%29
%28-tan%28C%29%29%281-tan%28A%29tan%28B%29%29=tan%28A%29%2Btan%28B%29
-tan%28C%29%2B%28-tan%28A%29tan%28B%29%29%28-tan%28C%29%29=tan%28A%29%2Btan%28B%29
-tan%28C%29%2Btan%28A%29tan%28B%29tan%28C%29=tan%28A%29%2Btan%28B%29
tan%28A%29tan%28B%29tan%28C%29=tan%28A%29%2Btan%28B%29%2Btan%28C%29
and as we were told that tan%28A%29%2Btan%28B%29%2Btan%28C%29=x , then
highlight%28tan%28A%29tan%28B%29tan%28C%29=x%29

Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
We want to find:

tan%28A%29tan%28B%29tan%28C%29

we know 

tan%28A%2BB%29=+%28tan%28A%29%2Btan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29

Since (A+B+C=180^o}}} 

A%2BB=180%5Eo-C

tan%28A%2BB%29=tan%28180%5Eo-C%29+=+-tan%28C%29

so substitute in

tan%28A%2BB%29=+%28tan%28A%29%2Btan%28B%29%29%2F%281-tanA%2AtanB%29

-tan%28C%29=+%28tan%28A%29%2Btan%28B%29%29%2F%281-tan%28A%29%2Atan%28B%29%29 

-tan%28C%29%2Btan%28A%29%2Atan%28B%29%2Atan%28C%29+=+tan%28A%29%2Btan%28B%29

tan%28A%29%2Atan%28B%29%2Atan%28C%29+=+tan%28A%29%2Btan%28B%29%2Btan%28C%29=x

So tan%28A%29%2Btan%28B%29%2Btan%28C%29=x also.

[It's interesting that the product and the sum of the tangents
of the angles of a non-right triangle are the same.  A right
triangle contains a 90° and tan(90°) is not defined.]

Edwin