Question

How do you solve the triangle given a = 39, b = 52, $ \angle C={{122}^{\circ }}$ ?

Answer 1

Hint: In order to do this question, need to use the law of cosines , which states that $ {{c}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\cos \left( C \right)$. Here, you can substitute all the values and find the side c. Then you can use the sine rule which states $ \dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$. From here, we can find the angles A and B. Then you can get the final triangle with all the sides and angles.

Complete step by step answer:
In order to do this question, need to use the law of cosines , which states that $ {{c}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\cos \left( C \right)$. Here, we will substitute all the values and find the side c.
$ \Rightarrow {{c}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\cos \left( C \right)$
We were given the values as a = 39, b = 52, $ \angle C={{122}^{\circ }}$
Substituting the above given values in the above equation of the cosine rule, we get the following:
$ \Rightarrow {{c}^{2}}={{39}^{2}}+{{52}^{2}}-2\times 39\times 52\cos \left( {{122}^{\circ }} \right)$
$ \Rightarrow {{c}^{2}}=1521+2704+2149.352$
$ \Rightarrow {{c}^{2}}=6374.352$
$ \Rightarrow c=79.839$
Now we can use the sine rule which states $ \dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$. From here, we can find the angles A and B. Then you can get the final triangle with all the sides and angles.
$ \Rightarrow \dfrac{\sin A}{39}=\dfrac{\sin 122}{79.839}$
$ \Rightarrow \sin A=\dfrac{\sin 122}{79.839}\times 39=0.414$
$ \Rightarrow A={{24.5}^{\circ }}$
$ \Rightarrow \dfrac{\sin B}{52}=\dfrac{\sin 122}{79.839}$
$ \Rightarrow \sin B=\dfrac{\sin 122}{79.839}\times 52=0.552$
$ \Rightarrow B={{30.5}^{\circ }}$

Therefore, we get , a = 39, b = 52, c = 79.839, $ \angle C={{122}^{\circ}} $ $\angle B={{30.5}^{\circ }}$, $ \angle A={{24.5}^{\circ }}$.

Note: In order to do this question, you need to know the cosine and sine laws. You can also find the angle B by subtracting the angles A and C from 180 as the sum of all the angles in a triangle equals to 180.