Question

The smallest angle of the triangle whose sides \[6 + \sqrt {12} ,\sqrt {48} \sqrt {24} \] are is
A. \[\frac{\pi }{3}\]
B. \[\frac{\pi }{4}\]
C. \[\frac{\pi }{6}\]
D. None of these

Answer 1

Hint: In order to solve problems involving right triangles, which involves solving for the length of a side of a triangle, you can use the Pythagorean theorem as follows: 
\[cosA + cosB = cosC\].
In this equation, \[A,{\rm{ }}B,\]and \[C\] are the lengths of the three sides of the triangle.
Formula use:
Cosine’s law formula is,
\[{a^2} = {b^2} + {c^2} - 2bc\cos A\]

Complete step-by-step solution:
In this case, if you have two angles with different lengths but smallest angles that are both \[180\]degrees (i.e., they each measure \[90\] degrees), then the larger of those two angles will be opposing to the smaller of those two angles. This is because the law of cosines dictates that the smallest angle is opposite to the shortest side.
The law of cosines applies since the smallest angle is opposite to the shortest side.
\[{a^2} = {b^2} + {c^2} - 2bc\cos A\]
On simplifying: \[\cos {\rm{A}} = \frac{{{{\rm{b}}^2} + {{\rm{c}}^2} - {{\rm{a}}^2}}}{{2{\rm{cb}}}}\]
By substituting the values, we currently hold
\[\cos A = \frac{{{{(6 + 2\sqrt 3 )}^2} + {{(4\sqrt 3 )}^2} - {{(2\sqrt 6 )}^2}}}{{2(6 + 2\sqrt 3 )(4\sqrt 3 )}}\]
\[ = \frac{{48 + 24\sqrt 3 + 48 - 24}}{{2(6 + 2\sqrt 3 )(4\sqrt 3 )}}\]
By expressing the numerator in terms of 2,
\[ = \frac{{36 + 12\sqrt 3 }}{{(6 + 2\sqrt 3 )(4\sqrt 3 )}}\]
Taking the numerator in terms of 2, \[ = \frac{{36 + 12\sqrt 3 }}{{(6 + 2\sqrt 3 )(4\sqrt 3 )}}\]
And lastly,
 \[\cos {\rm{A}} = \frac{{\sqrt 3}}{{ 2 }}\] \[ = \cos {30^\circ }\] \[ = \cos \frac{\pi }{6}\]
Hence, the smallest angle is opposite to the shortest side is \[\frac{\pi }{6}\]
So, Option C is correct.
Note:
Students fail to understand the Rule of Sins, the biggest mistake they make when solving this problem. This rule states that in order for two angles to be equal, they must also be opposite one another. Thus, angle \[A = B{\rm{ }}cosC\], where A and B are the original angles and C is the cosine of \[A/B\]. In other words, if you want to find \[C\], just divide both sides by \[cosC = \frac{\pi }{6}\].