First slide

Compound angles

Question

If $A > 0, B > 0$ and $A + B = Ï / 3$ then the maximum value of tan $A$ tan $B$ is

Moderate

A

$1 / \sqrt{3}$
B

$1 / 3$
C

$\sqrt{3}$
D

3

Solution

Let tan $A \tan B = x$
$\begin{array}{l} so \tan â¡ (A + B) = \tan â¡ (Ï / 3) = \sqrt{3} \\ â \frac{\tan â¡ A + \tan â¡ B}{1 â \tan â¡ A \tan â¡ B} = \sqrt{3} \\ â \sqrt{3} (1 â x) = \tan â¡ A + \tan â¡ B â¤ 2 \sqrt{\tan â¡ A \tan â¡ B} = 2 \sqrt{x} \\ â 3 (1 â x)^{2} â¤ 4 x \\ â 3 x^{2} â 10 x + 3 â¤ 0 \\ â (3 x â 1) (x â 3) â¤ 0 \\ â \frac{1}{3} â¤ x â¤ 3 \end{array}$

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