Chapter 6.2, Problem 64E

To determine

Whether the result is same while using the law of the Sine and then describe the advantage and the disadvantages of the both.

Answer to Problem 64E

No, the result is not same.

Explanation of Solution

The cosines law is given as:

  $\begin{array}{l}\cos A=\frac{b^2+c^2-a^2}{2bc}\\ \cos 20=\frac{{30}^2+c^2-{12}^2}{60c}\\ 56.38c=c^2+756\\ \text{Solving for }c:\\ c=34.41,21.97\end{array}$

Now using the sine law:

  $\begin{array}{l}\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\\ \text{Hence,}\\ \sin B=\frac{b}{a}\sin A\\ =\frac{30}{12}\sin 20\\ =0.855\\ A=58.76\end{array}$

Therefore, the angle C is evaluated as:

  $\begin{array}{l}C=180-\left(20+58.76\right)\\ =101.234\end{array}$

Hence, using the first and the last equality of the sine law as shown below:

  $\begin{array}{l}\frac{a}{\sin A}=\frac{c}{\sin C}\\ c=\frac{\sin C}{\sin A}a\\ c=\frac{\sin 101.234}{\sin 20}\left(12\right)\\ =34.41\end{array}$

Therefore, the correct answer is 34.41.

Sine law is better in comparison of the cosine law, as it will provide the unique result.