Question

$IfA+B+C=\pi$
lf $tanA:tanB:tanC=1:2:3$, then $\sin A:\sin B:\sin C=$

• A. $\sqrt{5}:3:2\sqrt{2}$
• B. $3:2\sqrt{2}:\sqrt{5}$
• C. $\sqrt{5}:2\sqrt{2}:3$
• D. $1:2:3$

Answer 1

The correct option is C $\sqrt{5}:2\sqrt{2}:3$

$tanA=K,tanB=2k,tanC=3K$
$sinA=\frac{k}{\sqrt{1+k^2}},sinB=\frac{2k}{\sqrt{1+4k^2}},sinC=\frac{3k}{\sqrt{1+9k^2}}$
$\because A+B+C=\pi$
$\therefore \sum tanA=\pi tanA$
$\therefore 6k=6k^3$
$\therefore k=1$
$\therefore sinA=\frac{1}{\sqrt{2}},sinB=\frac{2}{\sqrt{5}},sinC=\frac{3}{\sqrt{10}}$
$\therefore sinA:sinB:sinC=\sqrt{5}:2\sqrt{2}:3$