If the angles of a triangle are in the ratio $3 : 5 : 10$ then the ratio of smallest side to greatest side is

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$1 : 2 \sin 10^{0}$

$1 : \cos 10^{0}$

$1 : \sin 10^{0}$

$1 : 2 \cos 10^{0}$

answer is D.

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Detailed Solution

Given $A : B : C = 3 : 5 : 10$

Here $A$ is smallest angles and $C$ is longest angles

$A + B + C = 180 °$

So that $A = 30 °, C = 100 °$

Hence, the opposite sides are smalles and longest

$\begin{array}{rcl} \frac{a}{c} & = & \frac{2 R \sin A}{2 R \sin B} \\ = & \frac{\sin 30 °}{\sin 100 °} \\ = & \frac{1}{2 \cos 10 °} \end{array}$

Therefore, the ratio of shortest and longest side is $1 : 2 \cos 10 °$

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