Question

If $\tan^3 A + \tan^3 B + \tan^3 C = 3 \tan A \tan B \tan C$, then find the value of A + B + C.

• A. $\pi/2$
• B. $\pi$
• C. $2\pi$
• D. $3\pi/2$

Answer 1

Answer: (B)

$\pi$

Answer 2

Let $x = \tan A$, $y = \tan B$, and $z = \tan C$. The given equation is $x^3 + y^3 + z^3 = 3xyz$. We use the algebraic identity: $x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2 - xy - yz - zx)$

The given condition $x^3 + y^3 + z^3 = 3xyz$ implies $x^3 + y^3 + z^3 - 3xyz = 0$. Thus, we have: $(x+y+z)(x^2+y^2+z^2 - xy - yz - zx) = 0$

This equation holds if either:

1. $x+y+z = 0$
2. $x^2+y^2+z^2 - xy - yz - zx = 0$

The second condition can be rewritten as $\frac{1}{2}((x-y)^2 + (y-z)^2 + (z-x)^2) = 0$, which implies $x=y=z$.

So, we have two possibilities: a) $\tan A + \tan B + \tan C = 0$ b) $\tan A = \tan B = \tan C$

In the context of problems like this, where a unique value for $A+B+C$ is expected, it is usually implied that $A, B, C$ are the angles of a triangle. For the angles of a triangle, $A+B+C = \pi$.

If $A+B+C = \pi$, then it is a known trigonometric identity that: $\tan A + \tan B + \tan C = \tan A \tan B \tan C$ (provided none of the angles are $\pi/2$).

Let's examine our conditions in light of $A+B+C = \pi$:

Case 1: $\tan A + \tan B + \tan C = 0$ If $A+B+C=\pi$, then $\tan A + \tan B + \tan C = \tan A \tan B \tan C$. If $\tan A + \tan B + \tan C = 0$, then $\tan A \tan B \tan C$ must also be $0$. This implies at least one of $\tan A, \tan B, \tan C$ is $0$. If $\tan A = 0$, then $A = k\pi$ for some integer $k$. Since $A$ is an angle of a triangle, $A=0$. If $A=0$, then $B+C=\pi$. The condition $\tan A + \tan B + \tan C = 0$ becomes $0 + \tan B + \tan C = 0$, so $\tan B = -\tan C$. This is true if $B+C=\pi$. For example, $A=0, B=\pi/4, C=3\pi/4$. Here $A+B+C = \pi$.

Case 2: $\tan A = \tan B = \tan C$ If $A, B, C$ are angles of a triangle, and $\tan A = \tan B = \tan C$, then $A=B=C$. Since $A+B+C=\pi$, we have $3A = \pi$, which means $A = \pi/3$. So, $A=B=C=\pi/3$. In this case, $A+B+C = \pi$.

Both cases, under the assumption that $A, B, C$ are angles of a triangle, lead to $A+B+C = \pi$. If we do not assume $A, B, C$ are angles of a triangle, the sum $A+B+C$ can take infinitely many values. However, the question asks for "the value", implying a unique answer, which points to the standard interpretation of $A, B, C$ as angles of a triangle.

Therefore, the value of $A+B+C$ is $\pi$.