Question

The angles of a triangle area unit within the magnitude relation 3: 5: 10. Then, the magnitude relation of the tiniest aspect to the best aspect is
A) \[1:\sin {10^ \circ }\]
B) \[1:2\sin {10^ \circ }\]
C) \[1:\cos {10^ \circ }\]
D) \[1:2\cos {10^ \circ }\]

Answer 1

Hint: In this question, we have given the ratios of the angle of the triangle. First of all, we will determine the angle of the triangle. After that, we will apply the sine rule. And then we will select the correct answer from the given options.

Complete step by step solution: 
The sine rule is an equation that shows the relation between the sides of any triangles and the sine angles of the triangles. In other words, we can say that it is the relationship between the sides of the triangle and the sine angle. The side opposite the smallest angle is called the smallest side and the side opposite the greatest angle is called the greatest angle.
Let us consider that A, B, and C are the angles of the triangle and a, b, and c are the side of the triangle. And we know that the sum of the angle of the triangle is 180. Therefore we can write that
\[\begin{array}{*{20}{c}}
  {A + B + C}& = &{180}
\end{array}\] ……… (a)
According to the given question, we have the ratio of the angles of the triangles. So we will write.
\[\begin{array}{*{20}{c}}
  {A:B:C}& = &{3:5:10}
\end{array}\]
Now, we will get it.
\[\begin{array}{*{20}{c}}
  { \Rightarrow \dfrac{A}{3} = \dfrac{B}{5} = \dfrac{C}{{10}}}& = &y
\end{array}\]
On solving the above equation, we will get,
\[\begin{array}{*{20}{c}}
  { \Rightarrow A}& = &{3y}
\end{array}\] , \[\begin{array}{*{20}{c}}
  B& = &{5y}
\end{array}\] and \[\begin{array}{*{20}{c}}
  C& = &{10y}
\end{array}\]
Therefore, from equation (a), we will get,
\[ \Rightarrow \begin{array}{*{20}{c}}
  {3y + 5y + 10y}& = &{180}
\end{array}\]
\[ \Rightarrow \begin{array}{*{20}{c}}
  {18y}& = &{180}
\end{array}\]
 \[ \Rightarrow \begin{array}{*{20}{c}}
  y& = &{10}
\end{array}\]
Therefore, the angles will be
\[ \Rightarrow \begin{array}{*{20}{c}}
  A& = &{30}
\end{array}\], \[\begin{array}{*{20}{c}}
  B& = &{50}
\end{array}\] and \[\begin{array}{*{20}{c}}
  C& = &{100}
\end{array}\]
Now we know that the sine rule is,
\[\begin{array}{*{20}{c}}
  { \Rightarrow \dfrac{a}{{\sin A}} = \dfrac{b}{{\sin B}} = \dfrac{c}{{\sin C}}}& = &k
\end{array}\]
Therefore,
\[\begin{array}{*{20}{c}}
  { \Rightarrow \dfrac{a}{{\sin 30}} = \dfrac{b}{{\sin 50}} = \dfrac{c}{{\sin 100}}}& = &k
\end{array}\]
Here a is the smallest side and c is the greatest side of the triangle. So,
\[\begin{array}{*{20}{c}}
  { \Rightarrow \dfrac{a}{{\sin 30}}}& = &{\dfrac{c}{{\sin 100}}}
\end{array}\]
\[\begin{array}{*{20}{c}}
  { \Rightarrow \dfrac{a}{c}}& = &{\dfrac{{\sin 30}}{{\sin 100}}}
\end{array}\]
From the trigonometric table, we know that the value of is \[\dfrac{1}{2}\]. So, we can write
\[\begin{array}{*{20}{c}}
  { \Rightarrow \dfrac{a}{c}}& = &{\dfrac{1}{{2\sin (90 + 10)}}}
\end{array}\]
\[\begin{array}{*{20}{c}}
  { \Rightarrow a:c}& = &{1:2\cos 10}
\end{array}\]
Now the final answer is \[1:2\cos 10\].
So, the correct option is (D).

Note: Always remember that the opposite side of the largest angle is called the greatest side and the opposite side of the smallest angle is called the smallest side of the triangle.