Question

In a triangle ABC, $b=\sqrt{3},c=1$ and $\angle A={{30}^{0}}$ , then the largest angle of the triangle in degree is:
A. 135
B. 90
C. 60
D. 120

Answer 1

Hint: First we are going to use the cosine formula in triangles and using that we are going to find the value of the third side of the given triangle ABC and then by using the cosine formula again we will find all the three angles and then compare all of them to find the largest.

Complete step-by-step answer:

The cosine formula in a triangle is,
 Having side a, b, c

$\cos A=\dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}$
Now putting all the values we will find the value of ‘a’.
$\begin{align}
  & \cos \dfrac{\pi }{6}=\dfrac{{{\left( \sqrt{3} \right)}^{2}}+{{1}^{2}}-{{a}^{2}}}{2\sqrt{3}} \\
 & \dfrac{\sqrt{3}}{2}=\dfrac{4-{{a}^{2}}}{2\sqrt{3}} \\
 & 3=4-{{a}^{2}} \\
 & {{a}^{2}}=1 \\
\end{align}$
Therefore, a = 1.
Now we will use cosine formula again to find angle B,
$\begin{align}
  & \cos B=\dfrac{{{a}^{2}}+{{c}^{2}}-{{b}^{2}}}{2ac} \\
 & \cos B=\dfrac{1+1-{{\left( \sqrt{3} \right)}^{2}}}{2} \\
 & \cos B=\dfrac{2-3}{2} \\
 & \cos B=\dfrac{-1}{2} \\
 & \cos B=\cos \dfrac{2\pi }{3} \\
\end{align}$
Hence angle B = 120 degree.
Now using the fact that the sum of all three angles in a triangle is 180.
 $\begin{align}
  & \angle A+\angle B+\angle C=180 \\
 & 30+120+\angle C=180 \\
 & \angle C=30 \\
\end{align}$
Now we have found all the values of angles and from that we can compare and find the largest among them.
So, after comparing all the three angles we get,
The largest angle is 120.
Hence the correct answer is option (D).

Note: Here we have cosine formulas to solve this question, this formula is very useful to solve trigonometry questions and one can easily find the answer if we use this. There is a sine formula also which is also very important and both these formulas should be kept in mind and try to solve questions using them.