Chapter 6.2, Problem 63E

To determine

To explain:

law of cosine used for solving ambiguous case of given oblique triangle along with their advantages and disadvantages.

Answer to Problem 63E

law of cosine is used for solving given triangle

Explanation of Solution

Given information:

  $ABC$

Where, $a=12ft,b=30ft,A=20°$

It is given that,

  $a=12ft,b=30ft,A=20°$

By the law of cosine,

  $\begin{array}{l}{a}^2=b^2+c^2-2bc\cos A\\ c^{20}=a^2-b^2+2bc\cos A\end{array}$

Putting the values,

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

Thus, $c=34.47$ and $c=21.91$

Set of values given there are two triangles.

Thus there will be two values of angles $c=34.47$

By using law o f sine would give a same result as sine is positive for angles $0$ to $90°$ and $90°$ to $180°$ .

Advantages of using law of cosine:

Easier to choose the correct value to avoid ambiguous case.

Disadvantage of using law of cosine:

Computations and calculations are more.

Advantages of using law of sine:

Computations and calculations are less.

Disadvantages of using law of sine:

Not so easy to select the correct value and hence the chance of ambiguity is more.