A) \[\frac{1}{\sqrt{3}}\]
B) \[\sqrt{2}-1\]
C) \[\sqrt{3}\]
D) \[1\]
Correct Answer: A
Solution :
Given that, side of triangles are \[a=6+2\sqrt{3},\] \[b=4\sqrt{3}\] and \[c=\sqrt{24}\] Here, we observe that the side ‘c’ is small, hence the angles C is also small. Then, \[\cos \,\,C=\frac{{{a}^{2}}+{{b}^{2}}-{{c}^{2}}}{2ab}\] \[=\frac{{{(6+2\sqrt{3})}^{2}}+{{(4\sqrt{3})}^{2}}-{{(\sqrt{24})}^{2}}}{2(6+2\sqrt{3})(4\sqrt{3})}\] \[\Rightarrow \] \[\cos C=\frac{36+12+48-24+24\sqrt{3}}{16\sqrt{3}(3+\sqrt{3})}\] \[\Rightarrow \] \[\cos C=\frac{72+24\sqrt{3}}{16\sqrt{3}(3+\sqrt{3})}=\frac{24(3+\sqrt{3})}{16\sqrt{3}(3+\sqrt{3})}\] \[\Rightarrow \] \[\cos C=\frac{3}{2\sqrt{3}}=\frac{\sqrt{3}}{2}\] \[\Rightarrow \] \[\cos C=\cos {{30}^{o}}\Rightarrow \angle C={{30}^{o}}\] The smallest angle \[C={{30}^{o}}\] Hence, the tangent of smallest angle is \[\tan C=\tan {{30}^{o}}\] \[=\frac{1}{\sqrt{3}}\]
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