Question

What is the formula to find an angle when you have 3 sides of a triangle $a=13m,b=8.5m,c=9m$?

Answer 1

We know that if a, b and c are sides of a triangle then$\cos A=\dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}$, $\operatorname{cosB}=\dfrac{{{a}^{2}}+{{c}^{2}}-{{b}^{2}}}{2ac}$ and $\operatorname{cosC}=\dfrac{{{a}^{2}}+{{b}^{2}}-{{c}^{2}}}{2ab}$. By using this concept, the given concept can be solved.

Complete step by step solution:
From the question, it is clear that we have to find an angle when we are given 3 sides of a triangle $a=13m,b=8.5m,c=9m$.
We know that if a, b and c are sides of a triangle then$\cos A=\dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}$, $\operatorname{cosB}=\dfrac{{{a}^{2}}+{{c}^{2}}-{{b}^{2}}}{2ac}$ and $\operatorname{cosC}=\dfrac{{{a}^{2}}+{{b}^{2}}-{{c}^{2}}}{2ab}$.

So, by using the above formulae we can find the cosine of angle A, B and C of a triangle.
Let us assume

& a=13m.....(1) \\\ & b=8.5m.....(2) \\\ & c=9m.....(3) \\\ \end{aligned}$$ Let us assume $$\cos A=\dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}......(4)$$ Now let us substitute equation (1), equation (2) and equation (3) in equation (4), then we get $$\begin{aligned} & \Rightarrow \cos A=\dfrac{{{(8.5)}^{2}}+{{9}^{2}}-{{13}^{2}}}{2(8.5)(9)} \\\ & \Rightarrow \cos A=\dfrac{\dfrac{289}{4}+81-169}{153} \\\ & \Rightarrow \cos A=\dfrac{\dfrac{289}{4}-88}{153} \\\ & \Rightarrow \operatorname{cosA}=\dfrac{-7}{68} \\\ \end{aligned}$$ So, it is clear that $$\operatorname{cosA}=\dfrac{-7}{68}$$. Let us assume this as equation (5), then we get $$\operatorname{cosA}=\dfrac{-7}{68}.......(5)$$ From equation (5), we get $$\Rightarrow A={{\cos }^{-1}}\left( \dfrac{-7}{68} \right)$$ Let us assume this as equation this as equation (6). $$A={{\cos }^{-1}}\left( \dfrac{-7}{68} \right)......(6)$$ We know that $${{\cos }^{-1}}(-x)=\pi -{{\cos }^{-1}}x$$ Now let us assume the above concept to equation (6), then we get $$\Rightarrow A=\pi -{{\cos }^{-1}}\left( \dfrac{7}{68} \right)....(7)$$ From equation (7), it is clear that$$A=\pi -{{\cos }^{-1}}\left( \dfrac{7}{68} \right)$$. In this way we can find an angle when you have 3 sides of a triangle $$a=13m,b=8.5m,c=9m$$. **Note:** Students may have a misconception that if a, b and c are sides of a triangle then$$\cos A=\dfrac{{{b}^{2}}+{{c}^{2}}+{{a}^{2}}}{2bc}$$ but we know that if a, b and c are sides of a triangle then$$\cos A=\dfrac{{{b}^{2}}+{{c}^{2}}-{{a}^{2}}}{2bc}$$. Students should avoid these misconceptions to have a final answer not to get interrupted. Students should also avoid calculation mistakes while solving this problem.