Question

The sides of a triangle are $6+\sqrt {12} , \sqrt {48} $ and $\sqrt {24}$. The tangent of the smallest angle of the triangle is

• A. $\sqrt {3}$
• B. $1$
• C. $\frac {1}{\sqrt {3}}$
• D. $\sqrt {2}-1$

Answer 1

The Correct Option is C

Given that, side of triangles are \(a=6+2 \sqrt{3}, b=4 \sqrt{3} \text { and } c=\sqrt{24}\) Here, we observe that the side \(c\) is small, hence the angle \(C\) is also small. Then, \(\cos C=\frac{a^{2}+b^{2}-c^{2}}{2 a b}\) \(=\frac{(6+2 \sqrt{3})^{2}+(4 \sqrt{3})^{2}-(\sqrt{24})^{2}}{2(6+2 \sqrt{3})(4 \sqrt{3})}\) \(\Rightarrow \cos C=\frac{36+12+48-24+24 \sqrt{3}}{16 \sqrt{3}(3+\sqrt{3})}\) \(\Rightarrow \cos C=\frac{72+24 \sqrt{3}}{16 \sqrt{3}(3+\sqrt{3})}\) \(=\frac{24(3+\sqrt{3})}{16 \sqrt{3}(3+\sqrt{3})}\) \(\Rightarrow \cos C=\frac{3}{2 \sqrt{3}}=\frac{\sqrt{3}}{2}\) \(\Rightarrow \cos C=\cos 30^{\circ} \Rightarrow \angle C=30^{\circ}\) The smallest angle \(C=30^{\circ}\) Hence, the tangent of smallest angle is \(\tan C =\tan 30^{\circ}\) \(=\frac{1}{\sqrt{3}}\)

In a right-angled triangle, the tangent is the ratio between the adjacent and opposite sides of the angle that is being considered. The adjacent side denotes the side between the angle θ and the right angle, while the opposing denotes the side across from the reference angle θ. 

The tangent function has two important formulae. The ratio of the opposing side to the adjacent side of the angle under consideration is tan x in a right-angled triangle. The ratio of the sine function to the cosine function, which can be calculated using a unit circle, is another way to define the tangent function.

tan x = sin x/cos x

tan x = Opposite Side/Adjacent Side OR tan x= Perpendicular/Base