First slide

Compound angles

Question

If $\tan A, \tan B, \tan C$ satisfy the equation $3 \tan^{3} θ - 4 \tan^{2} θ + 3 \tan θ + 1 = 0,$ then $A + B + C =$

Moderate

A

$0$
B

$π / 2$
C

$3 π / 4$
D

$2 π$

Solution

$\begin{array}{l} S_{1} = \tan A + \tan B + \tan C = 4 / 3 \\ S_{2} = \sum \tan A \tan B = 1, \\ S_{3} = \tan A \tan B \tan C = - 1 / 3 \\ \tan (A + B + C) = \frac{S_{1} - S_{3}}{1 - S_{2}} \to \infty \\ \Rightarrow A + B + C = π / 2 \end{array}$

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