Chapter 6.2, Problem 63E

To determine

Tosolve:The triangle using law of cosine and compare the result by law of sine then describe advantage and disadvantage of each method.

Explanation of Solution

Given:

  $\begin{array}{l}a=12\text{ft}\\ b=30\text{ft}\\ A=20°\end{array}$

The value of $c$ by law of cosine is ,

  $\begin{array}{c}{c}^2=a^2-b^2+2bc\cos A\\ c^2={12}^2-{30}^2+2\left(12\right)\left(c\right)\cos 20°\\ c^2-56.38c+756=0\end{array}$

Solve the above quadratic expression.

  $c=34.47\text{and}\text{21}\text{.91}$ .

So, for the set of values given two triangles are possible.

By the law of sine it would give the same result, as sine is positive for angle $\text{0}\text{to}\text{90°}$ and $\text{90°}\text{to}\text{180°}$ .

Thus, there shall be two value of angle $C$ .

The advantage of law of cosine is it is easier to choose the correct value to avoid ambiguous case.

The disadvantage of law of cosine is that the computation and calculation are more.

The advantage of law of sine is that the computation and calculation are less.

The disadvantage of law of cosine is that it is not easy to choose the correct value and hence the ambiguity is more.